College

Multiply the polynomials.

\[
\left(4x^2 + 3x + 7\right)(8x - 5)
\]

A. \[ 32x^3 + 4x^2 + 41x + 35 \]

B. \[ 32x^3 - 44x^2 - 71x - 35 \]

C. \[ 32x^3 + 4x^2 + 41x - 35 \]

D. \[ 32x^3 - 4x^2 - 41x + 35 \]

Answer :

Sure, let's solve the polynomial multiplication step-by-step:

We need to multiply the two polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex].

Here's how to do it step by step:

1. Distribute each term in the first polynomial by each term in the second polynomial:

[tex]\[
(4x^2 + 3x + 7)(8x - 5) = 4x^2 \cdot 8x + 4x^2 \cdot (-5) + 3x \cdot 8x + 3x \cdot (-5) + 7 \cdot 8x + 7 \cdot (-5)
\][/tex]

2. Multiply the terms:

[tex]\[
4x^2 \cdot 8x = 32x^3
\][/tex]

[tex]\[
4x^2 \cdot (-5) = -20x^2
\][/tex]

[tex]\[
3x \cdot 8x = 24x^2
\][/tex]

[tex]\[
3x \cdot (-5) = -15x
\][/tex]

[tex]\[
7 \cdot 8x = 56x
\][/tex]

[tex]\[
7 \cdot (-5) = -35
\][/tex]

3. Combine all the products:

[tex]\[
32x^3 - 20x^2 + 24x^2 - 15x + 56x - 35
\][/tex]

4. Combine like terms:

Combine the [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] terms:

[tex]\[
-20x^2 + 24x^2 = 4x^2
\][/tex]

[tex]\[
-15x + 56x = 41x
\][/tex]

So, the combined polynomial is:

[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]

Therefore, the correct answer is:
[tex]\[
\boxed{C. \ 32x^3 + 4x^2 + 41x - 35}
\][/tex]