To find the expression equivalent to the given product of a binomial and a trinomial [tex]\( x^3 + 3x^2 - x + 2x^2 + 6x - 2 \)[/tex], we need to combine the like terms in the polynomial.
1. Identify and combine the like terms:
- Cubic term ([tex]\( x^3 \)[/tex]): There is only one [tex]\( x^3 \)[/tex] term, which is [tex]\( x^3 \)[/tex].
- Quadratic terms ([tex]\( x^2 \)[/tex]): Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
3x^2 + 2x^2 = 5x^2
\][/tex]
- Linear terms ([tex]\( x \)[/tex]): Combine the [tex]\( x \)[/tex] terms:
[tex]\[
-x + 6x = 5x
\][/tex]
- Constant term: There is only one constant term, which is [tex]\( -2 \)[/tex].
2. Write the simplified expression:
[tex]\[
x^3 + 5x^2 + 5x - 2
\][/tex]
Therefore, the expression equivalent to [tex]\( x^3 + 3x^2 - x + 2x^2 + 6x - 2 \)[/tex] after it has been fully simplified is:
[tex]\[
\boxed{x^3 + 5x^2 + 5x - 2}
\][/tex]
