Answer:
To multiply (x^2 + 3x + 4) and (3x^2 - 2x + 1), we need to distribute each term of the first polynomial to every term in the second polynomial:
(x^2 + 3x + 4)(3x^2 - 2x + 1)
= x^2 * (3x^2 - 2x + 1) + 3x * (3x^2 - 2x + 1) + 4 * (3x^2 - 2x + 1)
Now, let's simplify each term:
= 3x^4 - 2x^3 + x^2 + 9x^3 - 6x^2 + 3x + 12x^2 - 8x + 4
Combining like terms:
= 3x^4 + (-2x^3 + 9x^3) + (x^2 - 6x^2 + 12x^2) + (3x - 8x) + 4
= 3x^4 + 7x^3 + 7x^2 - 5x + 4
So, the product of (x^2 + 3x + 4) and (3x^2 - 2x + 1) is 3x^4 + 7x^3 + 7x^2 - 5x + 4.
Therefore, the correct answer is option C: 3x^4 + 7x^3 + 7x^2 - 5x + 4.