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Multiply the polynomials: [tex](8x^2 + 6x + 8)(6x - 5)[/tex]

A. [tex]48x^3 - 76x^2 + 18x - 40[/tex]
B. [tex]48x^3 - 4x^2 + 18x - 40[/tex]
C. [tex]48x^3 - 4x^2 + 18x + 40[/tex]
D. [tex]48x^3 - 4x^2 + 78x - 40[/tex]

Answer :

To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we will use the distributive property (also known as the FOIL method when applicable) to expand the expression.

Here's the step-by-step solution:

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

- First, distribute [tex]\(8x^2\)[/tex]:
[tex]\[
8x^2 \times 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \times (-5) = -40x^2
\][/tex]

- Next, distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \times 6x = 36x^2
\][/tex]
[tex]\[
6x \times (-5) = -30x
\][/tex]

- Finally, distribute [tex]\(8\)[/tex]:
[tex]\[
8 \times 6x = 48x
\][/tex]
[tex]\[
8 \times (-5) = -40
\][/tex]

2. Combine all the terms:

[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]

3. Combine like terms:

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

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

- The constant term remains [tex]\(-40\)[/tex].

4. Write the final expanded polynomial:

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

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

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