Answer :
To simplify the expression [tex]\((6x - 5)(2x^2 - 3x - 6)\)[/tex], we'll use the distributive property, often referred to as the FOIL method for binomials. This involves multiplying each term in the first expression by each term in the second expression.
Here's how you can do it step-by-step:
1. Multiply the first term of the first polynomial by each term of the second polynomial:
[tex]\[
6x \cdot 2x^2 = 12x^3
\][/tex]
[tex]\[
6x \cdot (-3x) = -18x^2
\][/tex]
[tex]\[
6x \cdot (-6) = -36x
\][/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
[tex]\[
-5 \cdot 2x^2 = -10x^2
\][/tex]
[tex]\[
-5 \cdot (-3x) = 15x
\][/tex]
[tex]\[
-5 \cdot (-6) = 30
\][/tex]
3. Combine all these results:
[tex]\[
12x^3 + (-18x^2) + (-36x) + (-10x^2) + 15x + 30
\][/tex]
4. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-36x + 15x = -21x\)[/tex]
5. Write the final simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Now, match this expression to the given options. The correct choice is:
- [tex]\( 12x^3 - 28x^2 - 21x + 30 \)[/tex]
So, the correct answer is the last option:
[tex]\[ \boxed{12x^3 - 28x^2 - 21x + 30} \][/tex]
Here's how you can do it step-by-step:
1. Multiply the first term of the first polynomial by each term of the second polynomial:
[tex]\[
6x \cdot 2x^2 = 12x^3
\][/tex]
[tex]\[
6x \cdot (-3x) = -18x^2
\][/tex]
[tex]\[
6x \cdot (-6) = -36x
\][/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
[tex]\[
-5 \cdot 2x^2 = -10x^2
\][/tex]
[tex]\[
-5 \cdot (-3x) = 15x
\][/tex]
[tex]\[
-5 \cdot (-6) = 30
\][/tex]
3. Combine all these results:
[tex]\[
12x^3 + (-18x^2) + (-36x) + (-10x^2) + 15x + 30
\][/tex]
4. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-18x^2 - 10x^2 = -28x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-36x + 15x = -21x\)[/tex]
5. Write the final simplified expression:
[tex]\[
12x^3 - 28x^2 - 21x + 30
\][/tex]
Now, match this expression to the given options. The correct choice is:
- [tex]\( 12x^3 - 28x^2 - 21x + 30 \)[/tex]
So, the correct answer is the last option:
[tex]\[ \boxed{12x^3 - 28x^2 - 21x + 30} \][/tex]
