To find the completely factored form of the polynomial [tex]\( x^3 + 3x^2 - 6x - 18 \)[/tex], we can follow a structured approach. Here is a detailed, step-by-step solution:
1. Identify the polynomial:
We are working with the polynomial:
[tex]\[
x^3 + 3x^2 - 6x - 18
\][/tex]
2. Factor the polynomial:
To factor the given polynomial, we attempt to express it as a product of simpler polynomials.
3. Consider factoring by grouping (if applicable):
Let’s split the polynomial into two groups:
[tex]\[
(x^3 + 3x^2) + (-6x - 18)
\][/tex]
Factor out the common factors in each group:
[tex]\[
x^2(x + 3) - 6(x + 3)
\][/tex]
Notice that [tex]\((x + 3)\)[/tex] is a common factor:
[tex]\[
(x^2 - 6)(x + 3)
\][/tex]
4. Write down the factored form:
Thus, the polynomial [tex]\( x^3 + 3x^2 - 6x - 18 \)[/tex] can be factored as:
[tex]\[
(x^2 - 6)(x + 3)
\][/tex]
5. Verify the factorization by expanding the factored form:
[tex]\[
(x + 3)(x^2 - 6) = x^3 + 3x^2 - 6x - 18
\][/tex]
This matches the original polynomial, confirming our factorization is correct.
6. Select the correct answer:
Based on the factorization, we see that the completely factored form of the polynomial is:
[tex]\[
(x^2 - 6)(x + 3)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B. (x^2 - 6)(x + 3)}
\][/tex]
