Alright, let's step through the multiplication of the two expressions:
Given:
[tex]\[
(3x - 6)(2x^2 - 7x + 1)
\][/tex]
We need to multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.
1. Distribute [tex]\(3x\)[/tex] to every 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]
2. Distribute [tex]\(-6\)[/tex] to every 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]
Now, combine all these results:
[tex]\[
6x^3 + (-21x^2) + 3x + (-12x^2) + 42x + (-6)
\][/tex]
Combine the like terms:
[tex]\[
6x^3 + (-21x^2 - 12x^2) + (3x + 42x) + (-6)
\][/tex]
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Thus, the correct product is:
[tex]\[
\boxed{6x^3 - 33x^2 + 45x - 6}
\][/tex]
