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Multiply:

[tex]
(x^4 + 1)(3x^2 + 9x + 2)
[/tex]

Options:

A. [tex] x^4 + 3x^2 + 9x + 3 [/tex]

B. [tex] 3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2 [/tex]

C. [tex] 3x^7 + 9x^6 + 2x^5 [/tex]

D. [tex] 3x^8 + 9x^4 + 2x^4 + 3x^2 + 9x + 2 [/tex]

Answer :

Sure! Let's multiply the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] step by step.

1. Distribute each term of [tex]\((x^4 + 1)\)[/tex] to [tex]\((3x^2 + 9x + 2)\)[/tex]:

- First, multiply [tex]\(x^4\)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:

[tex]\[
x^4 \cdot 3x^2 = 3x^6
\][/tex]

[tex]\[
x^4 \cdot 9x = 9x^5
\][/tex]

[tex]\[
x^4 \cdot 2 = 2x^4
\][/tex]

- Next, multiply [tex]\(1\)[/tex] by each term in [tex]\((3x^2 + 9x + 2)\)[/tex]:

[tex]\[
1 \cdot 3x^2 = 3x^2
\][/tex]

[tex]\[
1 \cdot 9x = 9x
\][/tex]

[tex]\[
1 \cdot 2 = 2
\][/tex]

2. Combine all the terms:

So, when we combine all these terms, the expanded expression is:

[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]

This is the result of multiplying [tex]\((x^4 + 1)\)[/tex] by [tex]\((3x^2 + 9x + 2)\)[/tex].

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