To determine the factors of the polynomial [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] by grouping, let's go through the process step by step.
1. Group the terms:
We have the polynomial [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex]. Let's group the terms in pairs:
[tex]\[
(x^3 + 5x^2) + (-6x - 30)
\][/tex]
2. Factor out the common factors in each group:
- In the first group [tex]\( x^3 + 5x^2 \)[/tex], the common factor is [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 5)
\][/tex]
- In the second group [tex]\( -6x - 30 \)[/tex], the common factor is [tex]\( -6 \)[/tex]:
[tex]\[
-6(x + 5)
\][/tex]
3. Rewrite the expression with the factored groups:
Combining these, we get:
[tex]\[
x^2(x + 5) - 6(x + 5)
\][/tex]
4. Factor out the common binomial factor [tex]\((x + 5)\)[/tex]:
Now, we see that [tex]\( (x + 5) \)[/tex] is a common factor in both terms:
[tex]\[
(x^2 - 6)(x + 5)
\][/tex]
So, the polynomial [tex]\( x^3 + 5x^2 - 6x - 30 \)[/tex] can be factored as [tex]\( (x^2 - 6)(x + 5) \)[/tex].
From the options provided, the correct grouping is:
[tex]\[
x^2(x + 5) - 6(x + 5)
\][/tex]
