An equivalent-expression question asks for a form that gives the same value for every input. When one number sits outside a short set of parentheses, use the distributive property: multiply it by each signed term inside. Desmos can check one input, but one match doesn't prove the expressions agree for every input. Watch especially for a negative factor multiplying a negative term.
Hints
- Hint 1
The distributive property means the factor outside parentheses multiplies both terms inside. Treat a subtraction like as before multiplying. What two products does that give you?
- Hint 2
Multiply the term with and the number without separately. A negative times a negative is positive, so what sign should the constant term, the term without , have?
Step-by-step
Distribute the negative factor
Step 1Check one input in Desmos
Let stand for in Desmos. Type , then ; Desmos prints . Choosing keeps the term with from disappearing. This is one check, not proof that a choice works for every value of .
- Step 2
Give each signed term its own product
The distributive property says the number outside parentheses multiplies every term inside. The second term is , not , so the outside factor multiplies both signed terms:
- Step 3
Multiply and match the expression
Multiply both products, remembering that two negatives make a positive:
So is the equivalent expression for every . Choice C.