Question

which polynomial represents the difference below?
$2x^2 + 7x + 6$
$\underline{\quad - \quad (3x^2 - x)}$
\\(\bigcirc\\) a. $-x^2 + 6x + 6$
\\(\bigcirc\\) b. $2x^2 + 4x + 6$
\\(\bigcirc\\) c. $2x^2 + 5x + 6$
\\(\bigcirc\\) d. $-x^2 + 8x + 6$

Explanation:

Step1: Distribute the negative sign

We have the expression \((2x^{2}+7x + 6)-(3x^{2}-x)\). Distribute the negative sign to the terms inside the second polynomial: \(2x^{2}+7x + 6-3x^{2}+x\).

Step2: Combine like terms

• For the \(x^{2}\) terms: \(2x^{2}-3x^{2}=-x^{2}\)
• For the \(x\) terms: \(7x + x = 8x\)
• The constant term remains \(6\)

Putting it all together, we get \(-x^{2}+8x + 6\)? Wait, no, wait. Wait, \(7x+x = 8x\)? Wait, no, wait the original first polynomial is \(2x^{2}+7x + 6\) and we subtract \((3x^{2}-x)\), so it's \(2x^{2}+7x + 6-3x^{2}+x\). Now combine \(x^{2}\) terms: \(2x^{2}-3x^{2}=-x^{2}\), \(x\) terms: \(7x + x=8x\), constant term \(6\). Wait, but let's check again. Wait, the options have D as \(-x^{2}+8x + 6\) and A as \(-x^{2}+6x + 6\). Wait, did I make a mistake? Wait, no, the original subtraction is \(2x^{2}+7x + 6-(3x^{2}-x)\). Let's do it step by step:

\(2x^{2}+7x + 6-3x^{2}+x=(2x^{2}-3x^{2})+(7x + x)+6=-x^{2}+8x + 6\). Wait, but let's check the options. Option D is \(-x^{2}+8x + 6\). Wait, but let me check the arithmetic again. \(7x - (-x)=7x + x = 8x\), yes. And \(2x^{2}-3x^{2}=-x^{2}\), constant term \(6\). So the result is \(-x^{2}+8x + 6\), which is option D? Wait, but let me check the problem again. Wait, the first polynomial is \(2x^{2}+7x + 6\), subtract \((3x^{2}-x)\). So:

\(2x^{2}+7x + 6-3x^{2}+x=(2 - 3)x^{2}+(7 + 1)x+6=-x^{2}+8x + 6\). So the correct answer is D.

Wait, but maybe I made a mistake in the sign. Wait, the subtraction is \((2x^{2}+7x + 6)-(3x^{2}-x)=2x^{2}+7x + 6-3x^{2}+x\). Combining like terms:

\(x^{2}\) terms: \(2x^{2}-3x^{2}=-x^{2}\)

\(x\) terms: \(7x + x = 8x\)

constant term: \(6\)

So the polynomial is \(-x^{2}+8x + 6\), which is option D.

Answer:

D. \(-x^{2}+8x + 6\)