High School

Multiply the polynomials.

[tex]\left(8x^2 + 6x + 8\right)(6x - 5)[/tex]

A. [tex]48x^3 - 4x^2 + 18x - 40[/tex]

B. [tex]48x^3 - 4x^2 + 18x + 40[/tex]

C. [tex]48x^3 - 4x^2 + 78x - 40[/tex]

D. [tex]48x^3 - 76x^2 + 18x - 40[/tex]

Answer :

To multiply the polynomials [tex]\((8x^2 + 6x + 8)\)[/tex] and [tex]\((6x - 5)\)[/tex], follow these steps:

1. Distribute each term of the first polynomial to each term of the second polynomial:

[tex]\[
(8x^2 + 6x + 8) \cdot (6x - 5) = 8x^2 \cdot (6x - 5) + 6x \cdot (6x - 5) + 8 \cdot (6x - 5)
\][/tex]

2. Multiply each pair of terms:

- Multiply [tex]\(8x^2\)[/tex] by each term in [tex]\(6x - 5\)[/tex]:
[tex]\[
8x^2 \cdot 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \cdot (-5) = -40x^2
\][/tex]

- Multiply [tex]\(6x\)[/tex] by each term in [tex]\(6x - 5\)[/tex]:
[tex]\[
6x \cdot 6x = 36x^2
\][/tex]
[tex]\[
6x \cdot (-5) = -30x
\][/tex]

- Multiply [tex]\(8\)[/tex] by each term in [tex]\(6x - 5\)[/tex]:
[tex]\[
8 \cdot 6x = 48x
\][/tex]
[tex]\[
8 \cdot (-5) = -40
\][/tex]

3. Combine all the terms:

Add all the results from step 2:
[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]

4. Combine like terms:

- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-40x^2 + 36x^2 = -4x^2
\][/tex]

- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 48x = 18x
\][/tex]

5. Write the final expression:

The resulting polynomial is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]

So, the correct answer is A. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].