GCSE Maths: Expanding double brackets (Higher)

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	3
   x	x2	3x
   2	2x	6

   First: x × x = x²
   Outer: x × 3 = 3x
   Inner: 2 × x = 2x
   Last: 2 × 3 = 6
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 3x + 2x + 6
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +3x, +2x
      add the coefficients: 3 + 2 = 5  so they give 5x
   number terms: +6  (only one of this type, so it stays as it is)
   Answer: x² + 5x + 6

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 30 and the answer gives 30. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	1
   x	x2	x
   5	5x	5

   First: x × x = x²
   Outer: x × 1 = x
   Inner: 5 × x = 5x
   Last: 5 × 1 = 5
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + x + 5x + 5
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +x, +5x
      add the coefficients: 1 + 5 = 6  so they give 6x
   number terms: +5  (only one of this type, so it stays as it is)
   Answer: x² + 6x + 5

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 32 and the answer gives 32. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	6
   x	x2	6x
   4	4x	24

   First: x × x = x²
   Outer: x × 6 = 6x
   Inner: 4 × x = 4x
   Last: 4 × 6 = 24
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 6x + 4x + 24
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +6x, +4x
      add the coefficients: 6 + 4 = 10  so they give 10x
   number terms: +24  (only one of this type, so it stays as it is)
   Answer: x² + 10x + 24

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 63 and the answer gives 63. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	y	2
   y	y2	2y
   7	7y	14

   First: y × y = y²
   Outer: y × 2 = 2y
   Inner: 7 × y = 7y
   Last: 7 × 2 = 14
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   y² + 2y + 7y + 14
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   y² terms: +y²  (only one of this type, so it stays as it is)
   y terms: +2y, +7y
      add the coefficients: 2 + 7 = 9  so they give 9y
   number terms: +14  (only one of this type, so it stays as it is)
   Answer: y² + 9y + 14

Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 50 and the answer gives 50. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	a	8
   a	a2	8a
   3	3a	24

   First: a × a = a²
   Outer: a × 8 = 8a
   Inner: 3 × a = 3a
   Last: 3 × 8 = 24
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   a² + 8a + 3a + 24
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   a² terms: +a²  (only one of this type, so it stays as it is)
   a terms: +8a, +3a
      add the coefficients: 8 + 3 = 11  so they give 11a
   number terms: +24  (only one of this type, so it stays as it is)
   Answer: a² + 11a + 24

Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives 66 and the answer gives 66. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	5
   x	x2	5x
   −2	−2x	−10

   First: x × x = x²
   Outer: x × 5 = 5x
   Inner: (−2) × x = −2x
   Last: (−2) × 5 = −10
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 5x − 2x − 10
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +5x, −2x
      add the coefficients: 5 − 2 = 3  so they give 3x
   number terms: −10  (only one of this type, so it stays as it is)
   Answer: x² + 3x − 10

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 8 and the answer gives 8. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−4
   x	x2	−4x
   3	3x	−12

   First: x × x = x²
   Outer: x × (−4) = −4x
   Inner: 3 × x = 3x
   Last: 3 × (−4) = −12
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 4x + 3x − 12
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: −4x, +3x
      add the coefficients: −4 + 3 = −1  so they give −x
   number terms: −12  (only one of this type, so it stays as it is)
   Answer: x² − x − 12

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −6 and the answer gives −6. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−6
   x	x2	−6x
   −1	−x	6

   First: x × x = x²
   Outer: x × (−6) = −6x
   Inner: (−1) × x = −x
   Last: (−1) × (−6) = 6
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 6x − x + 6
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: −6x, −x
      add the coefficients: −6 − 1 = −7  so they give −7x
   number terms: +6  (only one of this type, so it stays as it is)
   Answer: x² − 7x + 6

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives −6 and the answer gives −6. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	a	−2
   a	a2	−2a
   −5	−5a	10

   First: a × a = a²
   Outer: a × (−2) = −2a
   Inner: (−5) × a = −5a
   Last: (−5) × (−2) = 10
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   a² − 2a − 5a + 10
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   a² terms: +a²  (only one of this type, so it stays as it is)
   a terms: −2a, −5a
      add the coefficients: −2 − 5 = −7  so they give −7a
   number terms: +10  (only one of this type, so it stays as it is)
   Answer: a² − 7a + 10

Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives −2 and the answer gives −2. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	y	8
   y	y2	8y
   −3	−3y	−24

   First: y × y = y²
   Outer: y × 8 = 8y
   Inner: (−3) × y = −3y
   Last: (−3) × 8 = −24
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   y² + 8y − 3y − 24
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   y² terms: +y²  (only one of this type, so it stays as it is)
   y terms: +8y, −3y
      add the coefficients: 8 − 3 = 5  so they give 5y
   number terms: −24  (only one of this type, so it stays as it is)
   Answer: y² + 5y − 24

Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (x + 2)² = (x + 2)(x + 2)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	2
   x	x2	2x
   2	2x	4

   First: x × x = x²
   Outer: x × 2 = 2x
   Inner: 2 × x = 2x
   Last: 2 × 2 = 4
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 2x + 2x + 4
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +2x, +2x
      add the coefficients: 2 + 2 = 4  so they give 4x
   number terms: +4  (only one of this type, so it stays as it is)
   Answer: x² + 4x + 4

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 25 and the answer gives 25. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (x − 5)² = (x − 5)(x − 5)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−5
   x	x2	−5x
   −5	−5x	25

   First: x × x = x²
   Outer: x × (−5) = −5x
   Inner: (−5) × x = −5x
   Last: (−5) × (−5) = 25
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 5x − 5x + 25
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: −5x, −5x
      add the coefficients: −5 − 5 = −10  so they give −10x
   number terms: +25  (only one of this type, so it stays as it is)
   Answer: x² − 10x + 25

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (x + 10)² = (x + 10)(x + 10)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	10
   x	x2	10x
   10	10x	100

   First: x × x = x²
   Outer: x × 10 = 10x
   Inner: 10 × x = 10x
   Last: 10 × 10 = 100
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 10x + 10x + 100
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +10x, +10x
      add the coefficients: 10 + 10 = 20  so they give 20x
   number terms: +100  (only one of this type, so it stays as it is)
   Answer: x² + 20x + 100

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 169 and the answer gives 169. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (y − 1)² = (y − 1)(y − 1)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	y	−1
   y	y2	−y
   −1	−y	1

   First: y × y = y²
   Outer: y × (−1) = −y
   Inner: (−1) × y = −y
   Last: (−1) × (−1) = 1
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   y² − y − y + 1
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   y² terms: +y²  (only one of this type, so it stays as it is)
   y terms: −y, −y
      add the coefficients: −1 − 1 = −2  so they give −2y
   number terms: +1  (only one of this type, so it stays as it is)
   Answer: y² − 2y + 1

Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives 4 and the answer gives 4. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	3
   2x	2x2	6x
   1	x	3

   First: 2x × x = 2x²
   Outer: 2x × 3 = 6x
   Inner: 1 × x = x
   Last: 1 × 3 = 3
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   2x² + 6x + x + 3
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +2x²  (only one of this type, so it stays as it is)
   x terms: +6x, +x
      add the coefficients: 6 + 1 = 7  so they give 7x
   number terms: +3  (only one of this type, so it stays as it is)
   Answer: 2x² + 7x + 3

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 42 and the answer gives 42. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	4
   3x	3x2	12x
   −2	−2x	−8

   First: 3x × x = 3x²
   Outer: 3x × 4 = 12x
   Inner: (−2) × x = −2x
   Last: (−2) × 4 = −8
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   3x² + 12x − 2x − 8
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +3x²  (only one of this type, so it stays as it is)
   x terms: +12x, −2x
      add the coefficients: 12 − 2 = 10  so they give 10x
   number terms: −8  (only one of this type, so it stays as it is)
   Answer: 3x² + 10x − 8

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 49 and the answer gives 49. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	2x	5
   2x	4x2	10x
   3	6x	15

   First: 2x × 2x = 4x²
   Outer: 2x × 5 = 10x
   Inner: 3 × 2x = 6x
   Last: 3 × 5 = 15
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   4x² + 10x + 6x + 15
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +4x²  (only one of this type, so it stays as it is)
   x terms: +10x, +6x
      add the coefficients: 10 + 6 = 16  so they give 16x
   number terms: +15  (only one of this type, so it stays as it is)
   Answer: 4x² + 16x + 15

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 99 and the answer gives 99. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−2
   4x	4x2	−8x
   −1	−x	2

   First: 4x × x = 4x²
   Outer: 4x × (−2) = −8x
   Inner: (−1) × x = −x
   Last: (−1) × (−2) = 2
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   4x² − 8x − x + 2
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +4x²  (only one of this type, so it stays as it is)
   x terms: −8x, −x
      add the coefficients: −8 − 1 = −9  so they give −9x
   number terms: +2  (only one of this type, so it stays as it is)
   Answer: 4x² − 9x + 2

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−3
   x	x2	−3x
   3	3x	−9

   First: x × x = x²
   Outer: x × (−3) = −3x
   Inner: 3 × x = 3x
   Last: 3 × (−3) = −9
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 3x + 3x − 9
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: −3x, +3x
      add the coefficients: −3 + 3 = 0  so they cancel to give 0
   number terms: −9  (only one of this type, so it stays as it is)
   Answer: x² − 9

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	a	−8
   a	a2	−8a
   8	8a	−64

   First: a × a = a²
   Outer: a × (−8) = −8a
   Inner: 8 × a = 8a
   Last: 8 × (−8) = −64
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   a² − 8a + 8a − 64
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   a² terms: +a²  (only one of this type, so it stays as it is)
   a terms: −8a, +8a
      add the coefficients: −8 + 8 = 0  so they cancel to give 0
   number terms: −64  (only one of this type, so it stays as it is)
   Answer: a² − 64

Check. Test with numbers. Put a = 3 into the original and into your answer. The original gives −55 and the answer gives −55. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	2x	−5
   2x	4x2	−10x
   5	10x	−25

   First: 2x × 2x = 4x²
   Outer: 2x × (−5) = −10x
   Inner: 5 × 2x = 10x
   Last: 5 × (−5) = −25
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   4x² − 10x + 10x − 25
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +4x²  (only one of this type, so it stays as it is)
   x terms: −10x, +10x
      add the coefficients: −10 + 10 = 0  so they cancel to give 0
   number terms: −25  (only one of this type, so it stays as it is)
   Answer: 4x² − 25

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 11 and the answer gives 11. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	y	7
   y	y2	7y
   −7	−7y	−49

   First: y × y = y²
   Outer: y × 7 = 7y
   Inner: (−7) × y = −7y
   Last: (−7) × 7 = −49
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   y² + 7y − 7y − 49
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   y² terms: +y²  (only one of this type, so it stays as it is)
   y terms: +7y, −7y
      add the coefficients: 7 − 7 = 0  so they cancel to give 0
   number terms: −49  (only one of this type, so it stays as it is)
   Answer: y² − 49

Check. Test with numbers. Put y = 3 into the original and into your answer. The original gives −40 and the answer gives −40. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (x + 3)² = (x + 3)(x + 3)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	3
   x	x2	3x
   3	3x	9

   First: x × x = x²
   Outer: x × 3 = 3x
   Inner: 3 × x = 3x
   Last: 3 × 3 = 9
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² + 3x + 3x + 9
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +3x, +3x
      add the coefficients: 3 + 3 = 6  so they give 6x
   number terms: +9  (only one of this type, so it stays as it is)
   Answer: x² + 6x + 9

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 36 and the answer gives 36. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A term is one part of an expression, together with the sign in front of it (for example 3x or −5). Like terms have exactly the same letters with exactly the same powers (3x and 5x are like terms, 3x and 5x² are not). The coefficient is the number in front of the letters.

Method. Collect like terms: group the terms that have the same letters and powers, then add or subtract their coefficients (the numbers in front).

1. Write the terms out one by one, each with the sign in front of it. A term is one part of the expression. The sign in front of a term belongs to that term, so in 5x − 2y the terms are +5x and −2y. (A term with no sign in front is positive.)
   +x, +3, +x, +3, +x, +3, +x, +3
2. Sort the terms into groups of like terms. Like terms have the same letters with the same powers, so they are the same kind of thing. We can only add the same kind of thing: 3 apples + 2 apples = 5 apples, but 3 apples + 2 pears cannot be made into one number. Remember that x and x² are different kinds of term, and so are x and xy.
   x terms: +x, +x, +x, +x
      add the coefficients: 1 + 1 + 1 + 1 = 4  so they give 4x
   number terms: +3, +3, +3, +3
      add the coefficients: 3 + 3 + 3 + 3 = 12  so they give 12
3. Write the answer. Join the results of the groups. Put the term with the highest power first, and the number last.
   4x + 12

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 24 and the answer gives 24. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. The sign in front of a term goes with it. In 7x − 3x you are taking away 3 lots of x. Also, a letter on its own, such as x, has a coefficient of 1 (and −x has −1).

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−2
   x	x2	−2x
   7	7x	−14

   First: x × x = x²
   Outer: x × (−2) = −2x
   Inner: 7 × x = 7x
   Last: 7 × (−2) = −14
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 2x + 7x − 14
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: −2x, +7x
      add the coefficients: −2 + 7 = 5  so they give 5x
   number terms: −14  (only one of this type, so it stays as it is)
   Answer: x² + 5x − 14

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 10 and the answer gives 10. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Method. Substitute x = 5 into the expanded expression.

1. Write the expression. From part (a).
   x² + 5x − 14
2. Replace x by 5.
   5² + 5 × 5 − 14 = 25 + 25 − 14
3. Work it out.
   50 − 14 = 36 cm²

Check. Use the brackets directly: length = 5 + 7 = 12, width = 5 − 2 = 3, and 12 × 3 = 36. ✓

Method. Expand three brackets in two stages: expand the first two and tidy up, then multiply the result by the third bracket.

1. Write the question. Three brackets multiplied together. We can only multiply two things at a time, so we do it in two stages.
   (x + 1)(x + 2)(x + 3)
2. Stage 1: expand the first two brackets. Multiply every term of one bracket by every term of the other, exactly as for two brackets.

   ×	x	2
   x	x2	2x
   1	x	2

   x × x = x²
   x × 2 = 2x
   1 × x = x
   1 × 2 = 2
3. Collect the like terms from stage 1. We want a neat answer before the next multiplication. This reduces the amount of work in the next step.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +2x, +x
      add the coefficients: 2 + 1 = 3  so they give 3x
   number terms: +2  (only one of this type, so it stays as it is)
   So (x + 1)(x + 2) = x² + 3x + 2
4. Stage 2: multiply the answer from stage 1 by the third bracket. Now we have 3 terms times 2 terms, so 6 multiplications. Again every term must meet every term.

   ×	x	3
   x2	x3	3x2
   3x	3x2	9x
   2	2x	6

   x² × x = x³
   x² × 3 = 3x²
   3x × x = 3x²
   3x × 3 = 9x
   2 × x = 2x
   2 × 3 = 6
5. Collect the like terms. Add up the terms of the same kind.
   x³ + 3x² + 3x² + 9x + 2x + 6
   x³ terms: +x³  (only one of this type, so it stays as it is)
   x² terms: +3x², +3x²
      add the coefficients: 3 + 3 = 6  so they give 6x²
   x terms: +9x, +2x
      add the coefficients: 9 + 2 = 11  so they give 11x
   number terms: +6  (only one of this type, so it stays as it is)
   Answer: x³ + 6x² + 11x + 6

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 60 and the answer gives 60. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not try to do all three at once. Tidy the answer to stage 1 (collect like terms) before you multiply by the third bracket, or the algebra gets very long.

Method. Expand three brackets in two stages: expand the first two and tidy up, then multiply the result by the third bracket.

1. Write the question. Three brackets multiplied together. We can only multiply two things at a time, so we do it in two stages.
   (x − 1)(x + 2)(x + 4)
2. Stage 1: expand the first two brackets. Multiply every term of one bracket by every term of the other, exactly as for two brackets.

   ×	x	2
   x	x2	2x
   −1	−x	−2

   x × x = x²
   x × 2 = 2x
   (−1) × x = −x
   (−1) × 2 = −2
3. Collect the like terms from stage 1. We want a neat answer before the next multiplication. This reduces the amount of work in the next step.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +2x, −x
      add the coefficients: 2 − 1 = 1  so they give x
   number terms: −2  (only one of this type, so it stays as it is)
   So (x − 1)(x + 2) = x² + x − 2
4. Stage 2: multiply the answer from stage 1 by the third bracket. Now we have 3 terms times 2 terms, so 6 multiplications. Again every term must meet every term.

   ×	x	4
   x2	x3	4x2
   x	x2	4x
   −2	−2x	−8

   x² × x = x³
   x² × 4 = 4x²
   x × x = x²
   x × 4 = 4x
   (−2) × x = −2x
   (−2) × 4 = −8
5. Collect the like terms. Add up the terms of the same kind.
   x³ + 4x² + x² + 4x − 2x − 8
   x³ terms: +x³  (only one of this type, so it stays as it is)
   x² terms: +4x², +x²
      add the coefficients: 4 + 1 = 5  so they give 5x²
   x terms: +4x, −2x
      add the coefficients: 4 − 2 = 2  so they give 2x
   number terms: −8  (only one of this type, so it stays as it is)
   Answer: x³ + 5x² + 2x − 8

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 24 and the answer gives 24. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not try to do all three at once. Tidy the answer to stage 1 (collect like terms) before you multiply by the third bracket, or the algebra gets very long.

Method. Expand three brackets in two stages: expand the first two and tidy up, then multiply the result by the third bracket.

1. Write the question. Three brackets multiplied together. We can only multiply two things at a time, so we do it in two stages.
   (x + 2)(x + 2)(x − 1)
2. Stage 1: expand the first two brackets. Multiply every term of one bracket by every term of the other, exactly as for two brackets.

   ×	x	2
   x	x2	2x
   2	2x	4

   x × x = x²
   x × 2 = 2x
   2 × x = 2x
   2 × 2 = 4
3. Collect the like terms from stage 1. We want a neat answer before the next multiplication. This reduces the amount of work in the next step.
   x² terms: +x²  (only one of this type, so it stays as it is)
   x terms: +2x, +2x
      add the coefficients: 2 + 2 = 4  so they give 4x
   number terms: +4  (only one of this type, so it stays as it is)
   So (x + 2)(x + 2) = x² + 4x + 4
4. Stage 2: multiply the answer from stage 1 by the third bracket. Now we have 3 terms times 2 terms, so 6 multiplications. Again every term must meet every term.

   ×	x	−1
   x2	x3	−x2
   4x	4x2	−4x
   4	4x	−4

   x² × x = x³
   x² × (−1) = −x²
   4x × x = 4x²
   4x × (−1) = −4x
   4 × x = 4x
   4 × (−1) = −4
5. Collect the like terms. Add up the terms of the same kind.
   x³ − x² + 4x² − 4x + 4x − 4
   x³ terms: +x³  (only one of this type, so it stays as it is)
   x² terms: −x², +4x²
      add the coefficients: −1 + 4 = 3  so they give 3x²
   x terms: −4x, +4x
      add the coefficients: −4 + 4 = 0  so they cancel to give 0
   number terms: −4  (only one of this type, so it stays as it is)
   Answer: x³ + 3x² − 4

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 16 and the answer gives 16. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not try to do all three at once. Tidy the answer to stage 1 (collect like terms) before you multiply by the third bracket, or the algebra gets very long.

Method. Expand three brackets in two stages: expand the first two and tidy up, then multiply the result by the third bracket.

1. Write the question. Three brackets multiplied together. We can only multiply two things at a time, so we do it in two stages.
   (2x + 1)(x + 3)(x − 2)
2. Stage 1: expand the first two brackets. Multiply every term of one bracket by every term of the other, exactly as for two brackets.

   ×	x	3
   2x	2x2	6x
   1	x	3

   2x × x = 2x²
   2x × 3 = 6x
   1 × x = x
   1 × 3 = 3
3. Collect the like terms from stage 1. We want a neat answer before the next multiplication. This reduces the amount of work in the next step.
   x² terms: +2x²  (only one of this type, so it stays as it is)
   x terms: +6x, +x
      add the coefficients: 6 + 1 = 7  so they give 7x
   number terms: +3  (only one of this type, so it stays as it is)
   So (2x + 1)(x + 3) = 2x² + 7x + 3
4. Stage 2: multiply the answer from stage 1 by the third bracket. Now we have 3 terms times 2 terms, so 6 multiplications. Again every term must meet every term.

   ×	x	−2
   2x2	2x3	−4x2
   7x	7x2	−14x
   3	3x	−6

   2x² × x = 2x³
   2x² × (−2) = −4x²
   7x × x = 7x²
   7x × (−2) = −14x
   3 × x = 3x
   3 × (−2) = −6
5. Collect the like terms. Add up the terms of the same kind.
   2x³ − 4x² + 7x² − 14x + 3x − 6
   x³ terms: +2x³  (only one of this type, so it stays as it is)
   x² terms: −4x², +7x²
      add the coefficients: −4 + 7 = 3  so they give 3x²
   x terms: −14x, +3x
      add the coefficients: −14 + 3 = −11  so they give −11x
   number terms: −6  (only one of this type, so it stays as it is)
   Answer: 2x³ + 3x² − 11x − 6

Check. Test with numbers. Put x = 2 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not try to do all three at once. Tidy the answer to stage 1 (collect like terms) before you multiply by the third bracket, or the algebra gets very long.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 3 in the second, so we need 2 × 3 = 6 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. Multiply the numbers and letters, and take care with the signs.

   ×	x2	3x	−1
   x	x3	3x2	−x
   2	2x2	6x	−2

   x × x² = x³
   x × 3x = 3x²
   x × (−1) = −x
   2 × x² = 2x²
   2 × 3x = 6x
   2 × (−1) = −2
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x³ + 3x² − x + 2x² + 6x − 2
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x³ terms: +x³  (only one of this type, so it stays as it is)
   x² terms: +3x², +2x²
      add the coefficients: 3 + 2 = 5  so they give 5x²
   x terms: −x, +6x
      add the coefficients: −1 + 6 = 5  so they give 5x
   number terms: −2  (only one of this type, so it stays as it is)
   Answer: x³ + 5x² + 5x − 2

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 85 and the answer gives 85. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 3 in the second, so we need 2 × 3 = 6 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. Multiply the numbers and letters, and take care with the signs.

   ×	x2	−x	4
   2x	2x3	−2x2	8x
   −1	−x2	x	−4

   2x × x² = 2x³
   2x × (−x) = −2x²
   2x × 4 = 8x
   (−1) × x² = −x²
   (−1) × (−x) = x
   (−1) × 4 = −4
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   2x³ − 2x² + 8x − x² + x − 4
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x³ terms: +2x³  (only one of this type, so it stays as it is)
   x² terms: −2x², −x²
      add the coefficients: −2 − 1 = −3  so they give −3x²
   x terms: +8x, +x
      add the coefficients: 8 + 1 = 9  so they give 9x
   number terms: −4  (only one of this type, so it stays as it is)
   Answer: 2x³ − 3x² + 9x − 4

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 50 and the answer gives 50. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 3 in the second, so we need 2 × 3 = 6 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. Multiply the numbers and letters, and take care with the signs.

   ×	x2	3x	9
   x	x3	3x2	9x
   −3	−3x2	−9x	−27

   x × x² = x³
   x × 3x = 3x²
   x × 9 = 9x
   (−3) × x² = −3x²
   (−3) × 3x = −9x
   (−3) × 9 = −27
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x³ + 3x² + 9x − 3x² − 9x − 27
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x³ terms: +x³  (only one of this type, so it stays as it is)
   x² terms: +3x², −3x²
      add the coefficients: 3 − 3 = 0  so they cancel to give 0
   x terms: +9x, −9x
      add the coefficients: 9 − 9 = 0  so they cancel to give 0
   number terms: −27  (only one of this type, so it stays as it is)
   Answer: x³ − 27

Check. Test with numbers. Put x = 3 into the original and into your answer. The original gives 0 and the answer gives 0. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	x	−3y
   x	x2	−3xy
   2y	2xy	−6y2

   First: x × x = x²
   Outer: x × (−3y) = −3xy
   Inner: 2y × x = 2xy
   Last: 2y × (−3y) = −6y²
2. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   x² − 3xy + 2xy − 6y²
3. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   x² terms: +x²  (only one of this type, so it stays as it is)
   xy terms: −3xy, +2xy
      add the coefficients: −3 + 2 = −1  so they give −xy
   y² terms: −6y²  (only one of this type, so it stays as it is)
   Answer: x² − xy − 6y²

Check. Test with numbers. Put x = 3, y = 4 into the original and into your answer. The original gives −99 and the answer gives −99. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. There should be four pieces before you collect (2 × 2). Count them. A very common error is to multiply only the First and Last terms.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (2a + b)² = (2a + b)(2a + b)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	2a	b
   2a	4a2	2ab
   b	2ab	b2

   First: 2a × 2a = 4a²
   Outer: 2a × b = 2ab
   Inner: b × 2a = 2ab
   Last: b × b = b²
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   4a² + 2ab + 2ab + b²
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   a² terms: +4a²  (only one of this type, so it stays as it is)
   ab terms: +2ab, +2ab
      add the coefficients: 2 + 2 = 4  so they give 4ab
   b² terms: +b²  (only one of this type, so it stays as it is)
   Answer: 4a² + 4ab + b²

Check. Test with numbers. Put a = 3, b = 4 into the original and into your answer. The original gives 100 and the answer gives 100. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.

Key words. A binomial is an expression with two terms, like x + 3. Expanding two brackets means multiplying every term of one by every term of the other.

Method. To expand two brackets, multiply every term in the first bracket by every term in the second, then collect the like terms.

1. Write the squared bracket as two brackets. Squaring means multiplying by itself: (a + b)² means (a + b)(a + b). It does not mean you can just square each term.
   (3a − 2b)² = (3a − 2b)(3a − 2b)
2. Multiply every term in the first bracket by every term in the second bracket. There are 2 terms in the first bracket and 2 in the second, so we need 2 × 2 = 4 multiplications. A table (the grid method) keeps them organised: each box is the row term times the column term. (First, Outer, Inner, Last is a way of remembering the order.) Multiply the numbers and letters, and take care with the signs.

   ×	3a	−2b
   3a	9a2	−6ab
   −2b	−6ab	4b2

   First: 3a × 3a = 9a²
   Outer: 3a × (−2b) = −6ab
   Inner: (−2b) × 3a = −6ab
   Last: (−2b) × (−2b) = 4b²
3. Write down all the pieces. Add together the results of all the multiplications, each with its own sign.
   9a² − 6ab − 6ab + 4b²
4. Collect the like terms. The two middle pieces (the Outer and the Inner) are usually like terms. Add them. Pieces with different letters or powers stay separate.
   a² terms: +9a²  (only one of this type, so it stays as it is)
   ab terms: −6ab, −6ab
      add the coefficients: −6 − 6 = −12  so they give −12ab
   b² terms: +4b²  (only one of this type, so it stays as it is)
   Answer: 9a² − 12ab + 4b²

Check. Test with numbers. Put a = 3, b = 4 into the original and into your answer. The original gives 1 and the answer gives 1. They are the same, so the answer is right. ✓ (One test does not prove it, but a wrong answer almost never survives it.)

Watch out. Do not write (x + 3)² = x² + 9. The middle term is missing! (x + 3)(x + 3) has four pieces: x², 3x, 3x and 9.