To factor the polynomial [tex]\(3x^3 + 12x^2 + 2x + 8\)[/tex] completely using the grouping method, follow these steps:
1. Group the terms: We start by grouping the terms in pairs to make it easier to factor by grouping.
[tex]\[
3x^3 + 12x^2 + 2x + 8 = (3x^3 + 12x^2) + (2x + 8)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair of terms:
- For the first group, [tex]\(3x^3 + 12x^2\)[/tex], the GCF is [tex]\(3x^2\)[/tex]. So, we factor [tex]\(3x^2\)[/tex] out:
[tex]\[
3x^3 + 12x^2 = 3x^2(x + 4)
\][/tex]
- For the second group, [tex]\(2x + 8\)[/tex], the GCF is [tex]\(2\)[/tex]. So, we factor [tex]\(2\)[/tex] out:
[tex]\[
2x + 8 = 2(x + 4)
\][/tex]
Now our expression looks like this:
[tex]\[
3x^2(x + 4) + 2(x + 4)
\][/tex]
3. Factor out the common binomial factor: Notice that both terms now contain the common binomial factor [tex]\((x + 4)\)[/tex]. We factor [tex]\((x + 4)\)[/tex] out:
[tex]\[
3x^2(x + 4) + 2(x + 4) = (x + 4)(3x^2 + 2)
\][/tex]
So, the polynomial [tex]\(3x^3 + 12x^2 + 2x + 8\)[/tex] factors completely to [tex]\((x + 4)(3x^2 + 2)\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{C. (3x^2 + 2)(x + 4)}
\][/tex]
