A repeated piece in both terms is a common factor, even when one term has no number written in front of it. Rewrite a subtracted piece as negative one copy, then factor out the repeated piece and combine what remains. A Desmos value can catch a sign slip, but factoring shows the expressions match for every .
Hints
- Hint 1
An equivalent expression has the same value at every input. Testing one input in Desmos can catch a mismatch, but a match at that input doesn't prove equivalence. What algebraic move would prove it for every ?
- Hint 2
The last is being subtracted as a whole. Write it as so you can see the number multiplying that piece.
- Hint 3
Both terms contain the entire factor . When you pull a shared factor outside parentheses, each term leaves behind the part that multiplied it. What does the subtracted term leave behind?
Step-by-step
Factor the repeated piece
Step 1Get a value to catch sign slips
Type the given expression as , then type . Desmos shows . Choosing makes the terms vanish, so this is a quick check value. Matching at one input alone won't prove equivalence.
- Step 2
Show the hidden negative one
The last is subtracted as a whole. That means negative one whole copy of : .
- Step 3
Pull out the common factor
Factoring reverses distribution. Both terms have the whole factor , so pull it out and keep what multiplied it in each term: .
- Step 4
Combine what remains
Subtract from the constant inside the brackets: . The becomes , while stays unchanged. This factoring shows the product is equivalent for every . Choice D.