To multiply the polynomials [tex]\((x^2 + 3x + 4)\)[/tex] and [tex]\((3x^2 - 2x + 1)\)[/tex], we need to use the distributive property and multiply each term in the first polynomial by each term in the second polynomial. Here is the detailed step-by-step solution:
1. Distribute [tex]\(x^2\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
x^2 \cdot (3x^2 - 2x + 1) = (x^2 \cdot 3x^2) + (x^2 \cdot -2x) + (x^2 \cdot 1)
= 3x^4 - 2x^3 + x^2
\][/tex]
2. Distribute [tex]\(3x\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
3x \cdot (3x^2 - 2x + 1) = (3x \cdot 3x^2) + (3x \cdot -2x) + (3x \cdot 1)
= 9x^3 - 6x^2 + 3x
\][/tex]
3. Distribute [tex]\(4\)[/tex] from the first polynomial to each term in the second polynomial:
[tex]\[
4 \cdot (3x^2 - 2x + 1) = (4 \cdot 3x^2) + (4 \cdot -2x) + (4 \cdot 1)
= 12x^2 - 8x + 4
\][/tex]
4. Add all the terms together:
[tex]\[
3x^4 - 2x^3 + x^2 + 9x^3 - 6x^2 + 3x + 12x^2 - 8x + 4
\][/tex]
5. Combine like terms:
[tex]\[
3x^4 \quad \text{(only one \(x^4\) term)}
\][/tex]
[tex]\[
-2x^3 + 9x^3 = 7x^3 \quad \text{(combine \(x^3\) terms)}
\][/tex]
[tex]\[
x^2 - 6x^2 + 12x^2 = 7x^2 \quad \text{(combine \(x^2\) terms)}
\][/tex]
[tex]\[
3x - 8x = -5x \quad \text{(combine \(x\) terms)}
\][/tex]
[tex]\[
4 \quad \text{(only one constant term)}
\][/tex]
So, putting it all together, the product of the two polynomials is:
[tex]\[
3x^4 + 7x^3 + 7x^2 - 5x + 4
\][/tex]
Thus, the correct option is:
C. [tex]\(3x^4 + 7x^3 + 7x^2 - 5x + 4\)[/tex]
