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All courses Algebra II · A-APR.4 11 of 55
Prove polynomial identities and use them to solve or describe numerical relationships.

Verify a polynomial identity by expanding and comparing both sides

Problem
Verify \((x~+~2)^2~=~x^2~+~4x~+~4\) by expanding the left side.
Your answer
Show answer choicesHide answer choices Work through the mathematics first, then compare your reasoning.Select the answer that matches your work.
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Hint

Expand the left side \((x+2)^2\) using the binomial square pattern or direct multiplication.

\((x+2)^2 = (x+2)(x+2)\), so multiply each term and combine like terms.

Solution walkthrough

01

Rewrite the square as multiplication

\[(x+2)^2=(x+2)(x+2)\]

Squaring a binomial means multiplying it by an identical copy. Expanding this product will verify whether the proposed right side is identical for every x.

02

Distribute all four products

\[(x+2)(x+2)=x*x+x*2+2*x+2*2\]

Each term of the first binomial multiplies each term of the second. This explicitly includes both cross products x*2 and 2*x.

03

Evaluate and combine like terms

\[x^2+2x+2x+4=x^2+4x+4\]

The four products are x², 2x, 2x, and 4. Adding the two linear terms gives 4x, exactly the middle term on the proposed right side.

04

Compare and verify the identity

\[(x+2)^2=(x+2)(x+2)=x^2+2x+2x+4=x^2+4x+4,~\text{so}~\text{the}~\text{identity}~\text{is}~\text{verified}.\]

The expanded left side equals x²+4x+4 term for term. A check at x=1 gives 9 on both sides. Therefore the identity is verified exactly as stated in choice A.

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Another way

  1. Use the identity (a+b)²=a²+2ab+b² with a=x and b=2 to obtain x²+4x+4 directly.

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Common mistake

Do not square only the first and last terms. The two cross products total 4x, so (x+2)² is not x²+4.

Solution walkthrough video