To find the product of [tex]\((3x - 6)(2x^2 - 7x + 1)\)[/tex], let's perform the multiplication step by step. We need to distribute each term in the first binomial (3x - 6) to each term in the second polynomial (2x^2 - 7x + 1).
1. Distribute [tex]\(3x\)[/tex] to each term in [tex]\(2x^2 - 7x + 1\)[/tex]:
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
[tex]\[
3x \cdot (-7x) = -21x^2
\][/tex]
[tex]\[
3x \cdot 1 = 3x
\][/tex]
So, distributing [tex]\(3x\)[/tex] gives us: [tex]\(6x^3 - 21x^2 + 3x\)[/tex].
2. Distribute [tex]\(-6\)[/tex] to each term in [tex]\(2x^2 - 7x + 1\)[/tex]:
[tex]\[
-6 \cdot 2x^2 = -12x^2
\][/tex]
[tex]\[
-6 \cdot (-7x) = 42x
\][/tex]
[tex]\[
-6 \cdot 1 = -6
\][/tex]
So, distributing [tex]\(-6\)[/tex] gives us: [tex]\(-12x^2 + 42x - 6\)[/tex].
3. Combine the results from steps 1 and 2:
[tex]\[
6x^3 - 21x^2 + 3x + (-12x^2 + 42x - 6)
\][/tex]
Now combine like terms:
[tex]\[
6x^3 + (-21x^2 - 12x^2) + (3x + 42x) + (-6)
\][/tex]
Simplify the expression:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
So, the product of [tex]\((3x - 6)(2x^2 - 7x + 1)\)[/tex] is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Therefore, the correct answer among the given choices is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
