Answer :
Sure, let's multiply the polynomials step by step:
Given:
[tex]\[
(x^4 + 1)(3x^2 + 9x + 2)
\][/tex]
We will use the distributive property to multiply these polynomials. Let's distribute each term in the first polynomial across every term in the second polynomial.
### Step-by-Step Multiplication
1. 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]
2. 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]
Now, we add all these results together:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
### Final Answer
The product of the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
Given:
[tex]\[
(x^4 + 1)(3x^2 + 9x + 2)
\][/tex]
We will use the distributive property to multiply these polynomials. Let's distribute each term in the first polynomial across every term in the second polynomial.
### Step-by-Step Multiplication
1. 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]
2. 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]
Now, we add all these results together:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
### Final Answer
The product of the polynomials [tex]\((x^4 + 1)\)[/tex] and [tex]\((3x^2 + 9x + 2)\)[/tex] is:
[tex]\[
3x^6 + 9x^5 + 2x^4 + 3x^2 + 9x + 2
\][/tex]
