To multiply the polynomials [tex]\((3x^2 + 4x + 4)\)[/tex] and [tex]\((2x - 4)\)[/tex], we need to carefully distribute each term in the first polynomial by each term in the second polynomial. Here is the step-by-step solution:
1. Distribute [tex]\(3x^2\)[/tex] by each term in [tex]\(2x - 4\)[/tex]:
[tex]\[
3x^2 \cdot 2x = 6x^3
\][/tex]
[tex]\[
3x^2 \cdot (-4) = -12x^2
\][/tex]
2. Distribute [tex]\(4x\)[/tex] by each term in [tex]\(2x - 4\)[/tex]:
[tex]\[
4x \cdot 2x = 8x^2
\][/tex]
[tex]\[
4x \cdot (-4) = -16x
\][/tex]
3. Distribute [tex]\(4\)[/tex] by each term in [tex]\(2x - 4\)[/tex]:
[tex]\[
4 \cdot 2x = 8x
\][/tex]
[tex]\[
4 \cdot (-4) = -16
\][/tex]
4. Combine all the results:
[tex]\[
6x^3 - 12x^2 + 8x^2 - 16x + 8x - 16
\][/tex]
5. Combine like terms (terms with the same power of [tex]\(x\)[/tex]):
[tex]\[
6x^3 + (-12x^2 + 8x^2) + (-16x + 8x) - 16
\][/tex]
Simplifying this:
[tex]\[
6x^3 - 4x^2 - 8x - 16
\][/tex]
So, the product of [tex]\((3x^2 + 4x + 4)\)[/tex] and [tex]\((2x - 4)\)[/tex] is:
[tex]\[
6x^3 - 4x^2 - 8x - 16
\][/tex]
Therefore, the correct answer is:
B. [tex]\(6x^3 - 4x^2 - 8x - 16\)[/tex]
