Question

find the error cheyenne and nicolas are finding ((2x^2 - x) - (3x + 3x^2 - 2)). is either correct? explain your reasoning.
cheyenne
((2x^2 - x) - (3x + 3x^2 - 2) = (2x^2 - x) + (-3x + 3x^2 - 2))
(= 5x^2 - 4x - 2)
nicolas
((2x^2 - x) - (3x + 3x^2 - 2) = (2x^2 - x) + (-3x - 3x^2 + 2))
(= -x^2 - 4x + 2)
select choice select choice found the additive inverse correctly. all terms should have been multiplied by -1
need help?

Explanation:

Step1: Analyze Cheyenne's first step

When distributing the negative sign to the second polynomial, Cheyenne incorrectly kept $+3x^2$ instead of changing it to $-3x^2$.
Expression: $(2x^2 - x) - (3x + 3x^2 - 2) = (2x^2 - x) + (-3x - 3x^2 + 2)$ (correct additive inverse), but Cheyenne wrote $(2x^2 - x) + (-3x + 3x^2 - 2)$

Step2: Analyze Cheyenne's simplification

She combined like terms with the wrong sign on $3x^2$.
Expression: $2x^2 + 3x^2 - x - 3x - 2 = 5x^2 - 4x - 2$ (incorrect, as $3x^2$ should be $-3x^2$)

Step3: Analyze Nicolas's first step

Nicolas correctly found the additive inverse of the second polynomial by multiplying every term by $-1$.
Expression: $(2x^2 - x) + (-3x - 3x^2 + 2)$

Step4: Analyze Nicolas's simplification

He combined like terms correctly.
Expression: $2x^2 - 3x^2 - x - 3x + 2 = -x^2 - 4x + 2$
Note: Nicolas's final written answer has a typo ($-2$ instead of $+2$), but his first step of finding the additive inverse is correct, while Cheyenne's additive inverse was wrong.

Answer:

Cheyenne made an error when finding the additive inverse of the second polynomial: she did not flip the sign of $3x^2$ (it should be $-3x^2$ instead of $+3x^2$). Nicolas correctly found the additive inverse by multiplying all terms in the second polynomial by $-1$, though he had a typo in his final constant term (it should be $+2$, not $-2$). Overall, Nicolas's process is correct (aside from the final typo), while Cheyenne's initial step was wrong. The correct simplified form is $\boldsymbol{-x^2 - 4x + 2}$.