An equivalent expression has the same value for every input. When two terms contain the same parentheses, factoring lets you add the numbers outside them while leaving the parentheses unchanged. Add the outside multipliers, not the terms inside the shared parentheses. Desmos can check a rewrite at one input, but that check alone cannot prove equivalence.
Hints
- Hint 1
Both terms contain the entire factor . Factoring reverses distribution: what happens to the and outside that shared factor?
- Hint 2
For a quick check, set in Desmos and compare the original with your rewrite. Matching at one value is useful, but what algebra shows they match for every ?
Step-by-step
Factor the repeated parentheses
Step 1Pull out the shared factor
Both products contain the same factor, . Factoring reverses distribution: add the numbers multiplying that factor and write the factor once:
Don't add the two s inside the parentheses; each belongs to a whole product.
- Step 2
Check the rewrite at one input
Choose because it removes the -terms. Type , then and in Desmos. Both values print as , so the rewrite passes this check. One matching input isn't proof; factoring the shared factor is what makes the expressions equal for every .
- Step 3
Add the outside numbers
Add and , leaving the shared parentheses unchanged:
So the equivalent expression is . Choice C.