To simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex], let's break it down step by step using the distributive property (also known as the FOIL method for binomials):
1. Distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \cdot (2x^2 - 3x - 6) = 6x \cdot 2x^2 + 6x \cdot (-3x) + 6x \cdot (-6)
\][/tex]
2. Calculate each term individually:
- [tex]\(6x \cdot 2x^2 = 12x^3\)[/tex]
- [tex]\(6x \cdot (-3x) = -18x^2\)[/tex]
- [tex]\(6x \cdot (-6) = -36x\)[/tex]
3. Combine the results from distribution of [tex]\(6x\)[/tex]:
[tex]\[
12x^3 - 18x^2 - 36x
\][/tex]
4. Distribute [tex]\(-5\)[/tex]:
[tex]\[
-5 \cdot (2x^2 - 3x - 6) = -5 \cdot 2x^2 + (-5) \cdot (-3x) + (-5) \cdot (-6)
\][/tex]
5. Calculate each term individually:
- [tex]\(-5 \cdot 2x^2 = -10x^2\)[/tex]
- [tex]\(-5 \cdot (-3x) = 15x\)[/tex]
- [tex]\(-5 \cdot (-6) = 30\)[/tex]
6. Combine the results from distribution of [tex]\(-5\)[/tex]:
[tex]\[
-10x^2 + 15x + 30
\][/tex]
7. Add all the terms from both distributions together:
[tex]\[
(12x^3 - 18x^2 - 36x) + (-10x^2 + 15x + 30)
\][/tex]
8. Combine like terms:
[tex]\[
12x^3 + (-18x^2 - 10x^2) + (-36x + 15x) + 30
\][/tex]
Simplifying each group of like terms:
- [tex]\(12x^3\)[/tex]
- [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- [tex]\(-36x + 15x = -21x\)[/tex]
- [tex]\(30\)[/tex]
9. Combine all the simplified terms:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
So, the correct simplification of [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex] is:
[tex]\[
\boxed{12x^3 - 28x^2 - 21x + 30}
\][/tex]
The correct answer from the given choices is:
[tex]\(\boxed{12 x^3 - 28 x^2 - 21 x + 30}\)[/tex]
