Prove polynomial identities and use them to solve or describe numerical relationships.
Complete a missing term in a polynomial identity by expanding and comparing coefficients
Problem
Complete the identity \((x~+~5)^2~=~x^2~+~\text{\_\_}x~+~25\).
Big Picture
What this problem is really about
A missing coefficient can be found without guessing by expanding the structured side first. Map the two binomial terms into the square identity, isolate the middle term created by the two cross products, and then compare its coefficient with the blank. A direct multiplication provides a clean check that both cross products were counted.
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