To multiply the polynomials [tex]\((x^2 + 3x + 4)(3x^2 - 2x + 1)\)[/tex], we will perform the distribution (or FOIL) method. Let's multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Here are the steps:
1. Multiply [tex]\(x^2\)[/tex] by each term in [tex]\(3x^2 - 2x + 1\)[/tex]:
[tex]\[
x^2 \cdot 3x^2 = 3x^4
\][/tex]
[tex]\[
x^2 \cdot (-2x) = -2x^3
\][/tex]
[tex]\[
x^2 \cdot 1 = x^2
\][/tex]
2. Multiply [tex]\(3x\)[/tex] by each term in [tex]\(3x^2 - 2x + 1\)[/tex]:
[tex]\[
3x \cdot 3x^2 = 9x^3
\][/tex]
[tex]\[
3x \cdot (-2x) = -6x^2
\][/tex]
[tex]\[
3x \cdot 1 = 3x
\][/tex]
3. Multiply [tex]\(4\)[/tex] by each term in [tex]\(3x^2 - 2x + 1\)[/tex]:
[tex]\[
4 \cdot 3x^2 = 12x^2
\][/tex]
[tex]\[
4 \cdot (-2x) = -8x
\][/tex]
[tex]\[
4 \cdot 1 = 4
\][/tex]
Now, let's combine all these results:
[tex]\[
3x^4 + (-2x^3) + x^2 + 9x^3 + (-6x^2) + 3x + 12x^2 + (-8x) + 4
\][/tex]
Combine like terms:
[tex]\[
3x^4 + ( -2x^3 + 9x^3 ) + ( x^2 - 6x^2 + 12x^2 ) + ( 3x - 8x ) + 4
\][/tex]
Simplify the combined terms:
[tex]\[
3x^4 + 7x^3 + 7x^2 - 5x + 4
\][/tex]
Therefore, the product of the polynomials [tex]\((x^2 + 3x + 4)(3x^2 - 2x + 1)\)[/tex] is:
[tex]\[
3x^4 + 7x^3 + 7x^2 - 5x + 4
\][/tex]
So, the correct answer is [tex]\(D. 3x^4 + 7x^3 + 7x^2 - 5x + 4\)[/tex].
