To factor the polynomial [tex]\( x^3 + 2x^2 + 3x + 6 \)[/tex] completely using the grouping method, follow these detailed steps:
1. Group the terms: Split the polynomial into two groups.
[tex]\[
x^3 + 2x^2 + 3x + 6 = (x^3 + 2x^2) + (3x + 6)
\][/tex]
2. Factor out the greatest common factor (GCF) in each group:
- In the first group, [tex]\( x^3 + 2x^2 \)[/tex], the GCF is [tex]\( x^2 \)[/tex].
- In the second group, [tex]\( 3x + 6 \)[/tex], the GCF is [tex]\( 3 \)[/tex].
Factoring out the GCF from each group:
[tex]\[
x^2(x + 2) + 3(x + 2)
\][/tex]
3. Factor out the common binomial factor: Notice that both terms have a common factor of [tex]\( (x + 2) \)[/tex].
[tex]\[
x^2(x + 2) + 3(x + 2) = (x + 2)(x^2 + 3)
\][/tex]
Thus, the factored form of the polynomial [tex]\( x^3 + 2x^2 + 3x + 6 \)[/tex] is:
[tex]\[
(x + 2)(x^2 + 3)
\][/tex]
Therefore, the correct answer is:
D. [tex]\( (x^2 + 3)(x + 2) \)[/tex]
