To simplify the expression [tex]\((3x - 6)(2x^2 - 4x - 5)\)[/tex], we can follow these steps:
1. Distribute the terms:
- Distribute [tex]\(3x\)[/tex] to each term inside the parenthesis [tex]\(2x^2 - 4x - 5\)[/tex].
[tex]\[
3x \cdot (2x^2 - 4x - 5) = 3x \cdot 2x^2 + 3x \cdot (-4x) + 3x \cdot (-5)
\][/tex]
This expands to:
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
[tex]\[
3x \cdot (-4x) = -12x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
So, combining these terms:
[tex]\[
6x^3 - 12x^2 - 15x
\][/tex]
- Next, distribute [tex]\(-6\)[/tex] to each term inside the parenthesis [tex]\(2x^2 - 4x - 5\)[/tex].
[tex]\[
-6 \cdot (2x^2 - 4x - 5) = -6 \cdot 2x^2 + (-6) \cdot (-4x) + (-6) \cdot (-5)
\][/tex]
This expands to:
[tex]\[
-6 \cdot 2x^2 = -12x^2
\][/tex]
[tex]\[
-6 \cdot (-4x) = 24x
\][/tex]
[tex]\[
-6 \cdot (-5) = 30
\][/tex]
So, combining these terms:
[tex]\[
-12x^2 + 24x + 30
\][/tex]
2. Combine all the distributed terms:
[tex]\[
(6x^3 - 12x^2 - 15x) + (-12x^2 + 24x + 30)
\][/tex]
Combine like terms:
[tex]\[
6x^3 - 12x^2 - 12x^2 - 15x + 24x + 30
\][/tex]
Simplify by combining like terms:
[tex]\[
6x^3 - 24x^2 + 9x + 30
\][/tex]
Thus, the simplified form of the expression is:
[tex]\[
6x^3 - 24x^2 + 9x + 30
\][/tex]
So, the correct answer is:
[tex]\[6x^3 - 24x^2 + 9x + 30.\][/tex]
