To determine which expression is equivalent to the product given, we need to simplify the polynomial [tex]\( x^3 + 3x^2 - x + 2x^2 + 6x - 2 \)[/tex] by combining like terms. Here’s the step-by-step simplification process:
1. Identify and group like terms:
- [tex]\( x^3 \)[/tex] term: [tex]\( x^3 \)[/tex]
- [tex]\( x^2 \)[/tex] terms: [tex]\( 3x^2 \)[/tex] and [tex]\( 2x^2 \)[/tex]
- [tex]\( x \)[/tex] terms: [tex]\( -x \)[/tex] and [tex]\( 6x \)[/tex]
- Constant term: [tex]\( -2 \)[/tex]
2. Combine the coefficients of like terms:
- For [tex]\( x^3 \)[/tex]: There is only one [tex]\( x^3 \)[/tex] term, so the coefficient is 1. Thus, it remains [tex]\( x^3 \)[/tex].
- For [tex]\( x^2 \)[/tex]: Combine [tex]\( 3x^2 \)[/tex] and [tex]\( 2x^2 \)[/tex]. Adding the coefficients [tex]\( 3 \)[/tex] and [tex]\( 2 \)[/tex] yields [tex]\( 5 \)[/tex], so we have [tex]\( 5x^2 \)[/tex].
- For [tex]\( x \)[/tex]: Combine [tex]\( -x \)[/tex] and [tex]\( 6x \)[/tex]. Adding the coefficients [tex]\( -1 \)[/tex] and [tex]\( 6 \)[/tex] yields [tex]\( 5 \)[/tex], so we have [tex]\( 5x \)[/tex].
- The constant term remains [tex]\( -2 \)[/tex].
3. Write out the simplified polynomial:
The simplified form of the polynomial is:
[tex]\[
x^3 + 5x^2 + 5x - 2
\][/tex]
Now compare this simplified expression with the given options:
[tex]\[
\begin{align*}
\text{A. } & x^3 + 5x^2 + 5x - 2 \\
\text{B. } & x^3 + 2x^2 + 8x - 2 \\
\text{C. } & x^3 + 11x^2 - 2 \\
\text{D. } & x^3 + 10x^2 - 2 \\
\end{align*}
\][/tex]
Clearly, option A matches our simplified polynomial.
Therefore, the expression equivalent to the given product is:
[tex]\[
x^3 + 5x^2 + 5x - 2
\][/tex]
