Certainly! To factor the polynomial [tex]\(x^3 + 2x^2 + 12x + 24\)[/tex] using the grouping method, follow these steps:
1. Group the terms:
Split the polynomial into two groups that can be factored separately:
[tex]\[
x^3 + 2x^2 + 12x + 24 = (x^3 + 2x^2) + (12x + 24)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- From the first group [tex]\(x^3 + 2x^2\)[/tex], we can factor out [tex]\(x^2\)[/tex]:
[tex]\[
x^3 + 2x^2 = x^2(x + 2)
\][/tex]
- From the second group [tex]\(12x + 24\)[/tex], we can factor out [tex]\(12\)[/tex]:
[tex]\[
12x + 24 = 12(x + 2)
\][/tex]
3. Combine the factored groups:
After factoring each group, we get:
[tex]\[
x^3 + 2x^2 + 12x + 24 = x^2(x + 2) + 12(x + 2)
\][/tex]
4. Factor out the common binomial factor [tex]\((x + 2)\)[/tex]:
Notice that both terms have a common factor of [tex]\((x + 2)\)[/tex]:
[tex]\[
x^2(x + 2) + 12(x + 2) = (x^2 + 12)(x + 2)
\][/tex]
Thus, the polynomial [tex]\(x^3 + 2x^2 + 12x + 24\)[/tex] can be factored as:
[tex]\[
(x^2 + 12)(x + 2)
\][/tex]
Given the choices:
- A. [tex]\(\left(x^2+6\right)(x+2)\)[/tex]
- B. [tex]\(\left(x^2+2\right)(x+12)\)[/tex]
- C. [tex]\(\left(x^2+2\right)(x+6)\)[/tex]
- D. [tex]\(\left(x^2+12\right)(x+2)\)[/tex]
The correct option is:
[tex]\[
\boxed{(x^2 + 12)(x + 2)}
\][/tex]
Therefore, the correct choice is [tex]\( \text{D} \)[/tex].
