To simplify the expression [tex]\((3x - 6)(2x^2 - 4x - 5)\)[/tex], we need to perform polynomial multiplication by distributing each term in the first polynomial by each term in the second polynomial. Let's go through the steps in detail.
1. Distribute [tex]\(3x\)[/tex] to each term in [tex]\(2x^2 - 4x - 5\)[/tex]:
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
[tex]\[
3x \cdot (-4x) = -12x^2
\][/tex]
[tex]\[
3x \cdot (-5) = -15x
\][/tex]
2. Distribute [tex]\(-6\)[/tex] to each term in [tex]\(2x^2 - 4x - 5\)[/tex]:
[tex]\[
-6 \cdot 2x^2 = -12x^2
\][/tex]
[tex]\[
-6 \cdot (-4x) = 24x
\][/tex]
[tex]\[
-6 \cdot (-5) = 30
\][/tex]
3. Combine all the terms obtained from the above distributions:
[tex]\[
6x^3 - 12x^2 - 15x - 12x^2 + 24x + 30
\][/tex]
4. Combine and simplify like terms:
[tex]\[
6x^3 + (-12x^2 - 12x^2) + (-15x + 24x) + 30
\][/tex]
[tex]\[
6x^3 - 24x^2 + 9x + 30
\][/tex]
Therefore, the correct simplification of the expression [tex]\((3x - 6)(2x^2 - 4x - 5)\)[/tex] is:
[tex]\[6x^3 - 24x^2 + 9x + 30\][/tex]
So, the correct answer is:
[tex]\[ \boxed{6x^3 - 24x^2 + 9x + 30} \][/tex]
