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Multiply the polynomials.

[tex]\left(7x^2 + 5x + 7\right)(4x - 6)[/tex]

A. [tex]28x^3 - 22x^2 - 2x - 42[/tex]
B. [tex]28x^3 - 22x^2 - 58x - 42[/tex]
C. [tex]28x^3 - 22x^2 - 2x + 42[/tex]
D. [tex]28x^3 - 62x^2 - 2x - 42[/tex]

Answer :

Sure! Let's go through the steps of multiplying the polynomials:

[tex]\[
(7x^2 + 5x + 7)(4x - 6)
\][/tex]

We'll multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.

1. Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex] and [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \cdot 4x = 28x^3
\][/tex]
[tex]\[
7x^2 \cdot (-6) = -42x^2
\][/tex]

2. Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex] and [tex]\(-6\)[/tex]:
[tex]\[
5x \cdot 4x = 20x^2
\][/tex]
[tex]\[
5x \cdot (-6) = -30x
\][/tex]

3. Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex] and [tex]\(-6\)[/tex]:
[tex]\[
7 \cdot 4x = 28x
\][/tex]
[tex]\[
7 \cdot (-6) = -42
\][/tex]

Now, let's combine all these terms:

[tex]\[
28x^3 + (-42x^2 + 20x^2) + (-30x + 28x) - 42
\][/tex]

Simplify by combining like terms:

1. Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]

2. Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]

So, the resulting polynomial is:

[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]

Thus, the correct answer is:

[tex]\[
\boxed{A. \ 28x^3 - 22x^2 - 2x - 42}
\][/tex]

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