High School

Multiply:

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

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 :

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

Step 1: Distribute [tex]\(x^4\)[/tex] from the first expression
- Multiply [tex]\(x^4\)[/tex] by each term in the second expression:

[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]

Step 2: Distribute [tex]\(1\)[/tex] from the first expression
- Multiply [tex]\(1\)[/tex] by each term in the second expression:

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

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

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

Step 3: Combine all the products
- List all the terms obtained from both distributive steps:

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

This is the expanded form of the product [tex]\((x^4 + 1)(3x^2 + 9x + 2)\)[/tex].

So, the final result is:

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

That's the solution!