Sure, let's factor the polynomial expression [tex]\(8x^3 - 8x^2 - 30x\)[/tex].
### Step-by-Step Solution:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of all the terms in the polynomial. The GCF of [tex]\(8x^3\)[/tex], [tex]\(-8x^2\)[/tex], and [tex]\(-30x\)[/tex] is [tex]\(2x\)[/tex].
2. Factor out the GCF:
Factor [tex]\(2x\)[/tex] out from each term:
[tex]\[
8x^3 - 8x^2 - 30x = 2x(4x^2 - 4x - 15)
\][/tex]
3. Factor the Quadratic Expression:
Next, we need to factor the quadratic expression [tex]\(4x^2 - 4x - 15\)[/tex]. We look for two numbers whose product is [tex]\((4 \cdot -15) = -60\)[/tex] and whose sum is [tex]\(-4\)[/tex].
The pair of numbers [tex]\((6 \text{ and } -10)\)[/tex] fit this requirement because [tex]\(6 \cdot (-10) = -60\)[/tex] and [tex]\(6 + (-10) = -4\)[/tex].
4. Rewrite the Quadratic Expression:
Rewrite [tex]\(4x^2 - 4x - 15\)[/tex] using these numbers:
[tex]\[
4x^2 - 4x - 15 = 4x^2 + 6x - 10x - 15
\][/tex]
5. Group the Terms:
Group terms to factor by grouping:
[tex]\[
= (4x^2 + 6x) + (-10x - 15)
\][/tex]
6. Factor by Grouping:
Factor out the common factors in each group:
[tex]\[
= 2x(2x + 3) - 5(2x + 3)
\][/tex]
7. Combine the Factors:
Notice that [tex]\((2x + 3)\)[/tex] is a common factor:
[tex]\[
= (2x + 3)(2x - 5)
\][/tex]
8. Combine All Factors:
Bring back the GCF we factored out at the beginning ([tex]\(2x\)[/tex]):
[tex]\[
8x^3 - 8x^2 - 30x = 2x(2x + 3)(2x - 5)
\][/tex]
So, the factored form of the expression [tex]\(8x^3 - 8x^2 - 30x\)[/tex] is:
[tex]\[
\boxed{2x(2x + 3)(2x - 5)}
\][/tex]
Thus, the correct entries from the drop-down menus should be:
- 2x
- (2x + 3)
- (2x - 5)
So, [tex]\(8x^3 - 8x^2 - 30x = \boxed{2x(2x + 3)(2x - 5)}\)[/tex].
