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Quadratics: Multiplying Polynomials

A free College Algebra lesson from the “Quadratics: Multiplying and Factoring” unit, with a worked example and practice problems including step-by-step solutions.

Quadratic products come from multiplying binomials, such as (x + 2)(x + 5). Each term in one factor must multiply each term in the other factor. This process explains where quadratic trinomials come from: the first terms create the x^2 term, the outer and inner products combine to create the middle term, and the constants create the final term. This matters because expanding and factoring are inverse skills. When practicing, distribute systematically and combine like terms. A common mistake is multiplying only the first and last terms and forgetting the middle products.

What you'll learn

Why it matters: Area products, package dimensions, and compound expressions all make the four products in a binomial multiplication visible. The area grid keeps students from dropping the middle terms, which is the structural bridge to factoring.

Worked example

Problem. Expand (x + 4)(x - 7).

  1. Multiply first, outer, inner, last.
  2. x^2 - 7x + 4x - 28.
  3. Combine like terms to get x^2 - 3x - 28.

Answer: x^2 - 3x - 28

Practice problems

1. Expand (x + 2)(x + 6).

Choices: x^2 + 8x + 12 · x^2 + 12 · x^2 + 4x + 12 · 2x + 8

Show solution
  1. Warm-up: First identify exactly what the question is asking: Expand (x + 2)(x + 6).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms add to 8x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 + 8x + 12

2. Expand (x - 5)^2.

Choices: x^2 - 10x + 25 · x^2 - 25 · x^2 + 10x + 25 · x^2 - 5x + 25

Show solution
  1. Core Practice: First identify exactly what the question is asking: Expand (x - 5)^2.
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Square both terms and double the product.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 - 10x + 25

3. Expand (2x + 3)(x - 4).

Choices: 2x^2 - 5x - 12 · 2x^2 - 8x + 3 · 2x^2 + 11x - 12 · 3x^2 - x - 12

Show solution
  1. Challenge: First identify exactly what the question is asking: Expand (2x + 3)(x - 4).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are -8x and 3x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 2x^2 - 5x - 12

4. Expand (x + 7)(x - 1).

Choices: x^2 + 6x - 7 · x^2 - 8x - 7 · x^2 + 7x - 1 · x^2 - 7

Show solution
  1. Core Practice: First identify exactly what the question is asking: Expand (x + 7)(x - 1).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are -x and 7x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x^2 + 6x - 7

5. Expand (3x - 2)(x + 5).

Choices: 3x^2 + 13x - 10 · 3x^2 + 15x - 10 · 3x^2 - 7x - 10 · 4x^2 + 3x - 10

Show solution
  1. Core Practice: First identify exactly what the question is asking: Expand (3x - 2)(x + 5).
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The middle terms are 15x and -2x.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3x^2 + 13x - 10

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