To find the product of [tex]\((3x - 4)(2x^2 + 2x - 1)\)[/tex], we can follow these steps:
1. Expand the expression by distributing each term in the first polynomial to every term in the second polynomial.
2. Combine like terms.
Let's start with the expansion:
[tex]\[
(3x - 4)(2x^2 + 2x - 1)
\][/tex]
First, distribute [tex]\(3x\)[/tex] to each term in [tex]\(2x^2 + 2x - 1\)[/tex]:
[tex]\[
3x(2x^2) + 3x(2x) + 3x(-1)
\][/tex]
[tex]\[
= 6x^3 + 6x^2 - 3x
\][/tex]
Next, distribute [tex]\(-4\)[/tex] to each term in [tex]\(2x^2 + 2x - 1\)[/tex]:
[tex]\[
-4(2x^2) + (-4)(2x) + (-4)(-1)
\][/tex]
[tex]\[
= -8x^2 - 8x + 4
\][/tex]
Now, combine these results:
[tex]\[
6x^3 + 6x^2 - 3x - 8x^2 - 8x + 4
\][/tex]
Next, combine like terms:
[tex]\[
6x^3 + (6x^2 - 8x^2) + (-3x - 8x) + 4
\][/tex]
[tex]\[
6x^3 + (-2x^2) + (-11x) + 4
\][/tex]
The final simplified expression is:
[tex]\[
6x^3 - 2x^2 - 11x + 4
\][/tex]
So, among the given options, this corresponds to:
[tex]\[
\boxed{6x^3 - 2x^2 - 11x + 4}
\][/tex]
