Sure, let's find the product of the given polynomials step by step:
We start with the polynomials:
[tex]\[
(5x + 8 - 6x)(4 + 2x - 7)
\][/tex]
1. Simplify the first polynomial:
[tex]\[
5x + 8 - 6x
\][/tex]
Combine like terms:
[tex]\[
(5 - 6)x + 8 = -x + 8
\][/tex]
So the simplified first polynomial is:
[tex]\[
-x + 8
\][/tex]
2. Expand the product of the two polynomials:
[tex]\[
(-x + 8)(4 + 2x - 7)
\][/tex]
First, distribute [tex]\(-x\)[/tex]:
[tex]\[
-x \cdot 4 + (-x) \cdot 2x + (-x) \cdot (-7)
\][/tex]
[tex]\[
= -4x - 2x^2 + 7x
\][/tex]
Then, distribute [tex]\(8\)[/tex]:
[tex]\[
8 \cdot 4 + 8 \cdot 2x + 8 \cdot (-7)
\][/tex]
[tex]\[
= 32 + 16x - 56
\][/tex]
Now, add these results together:
[tex]\[
-4x - 2x^2 + 7x + 32 + 16x - 56
\][/tex]
3. Combine like terms:
[tex]\[
-2x^2 + (-4x + 7x + 16x) + (32 - 56)
\][/tex]
[tex]\[
-2x^2 + 19x - 24
\][/tex]
So, the product of the polynomials is:
[tex]\[
-2x^2 + 19x - 24
\][/tex]
Comparing this to the given choices:
A. [tex]\(2x^2 + 13x - 24\)[/tex]
B. [tex]\(-2x^2 + 19x - 24\)[/tex]
C. [tex]\(-2x^2 - 24x + 19\)[/tex]
D. [tex]\(2x^2 + 19x + 24\)[/tex]
The correct answer is:
[tex]\[
\boxed{-2x^2 + 19x - 24}
\][/tex] which matches choice B.
