Certainly! Let's use the grouping method to factor the polynomial [tex]\(2x^3 + 6x^2 - 7x - 21\)[/tex]. Here’s a step-by-step solution:
1. Rewrite the polynomial in two groups:
[tex]\[2x^3 + 6x^2 - 7x - 21\][/tex]
Let's divide this polynomial into two parts:
[tex]\[ (2x^3 + 6x^2) + (-7x - 21) \][/tex]
2. Factor out the greatest common factor (GCF) in each group:
- From the first group [tex]\(2x^3 + 6x^2\)[/tex], the GCF is [tex]\(2x^2\)[/tex]:
[tex]\[ 2x^2(x + 3) \][/tex]
- From the second group [tex]\(-7x - 21\)[/tex], the GCF is [tex]\(-7\)[/tex]:
[tex]\[ -7(x + 3) \][/tex]
Now the polynomial looks like:
[tex]\[ 2x^2(x + 3) - 7(x + 3) \][/tex]
3. Factor out the common binomial factor:
We see that [tex]\((x + 3)\)[/tex] is a common factor in both terms:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]
Therefore, the factored form of the polynomial [tex]\(2x^3 + 6x^2 - 7x - 21\)[/tex] is:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]
4. Compare with the given options:
We can observe that this matches option B:
[tex]\[ (x + 3)(2x^2 - 7) \][/tex]
So, the correct answer is:
[tex]\[ \boxed{(x + 3)\left(2x^2 - 7\right)} \][/tex]
Thus, the correct option is [tex]\( \text{B} \)[/tex].
