To find the product of the expressions [tex]\((3x - 6)\)[/tex] and [tex]\((2x^2 - 7x + 1)\)[/tex], we'll follow these steps:
1. Distribute [tex]\((3x - 6)\)[/tex] across each term in the polynomial [tex]\((2x^2 - 7x + 1)\)[/tex].
First, distribute [tex]\(3x\)[/tex]:
[tex]\[
3x \cdot (2x^2) + 3x \cdot (-7x) + 3x \cdot 1
\][/tex]
This gives us:
[tex]\[
6x^3 - 21x^2 + 3x
\][/tex]
Next, distribute [tex]\(-6\)[/tex]:
[tex]\[
-6 \cdot (2x^2) + -6 \cdot (-7x) + -6 \cdot 1
\][/tex]
This gives us:
[tex]\[
-12x^2 + 42x - 6
\][/tex]
2. Combine all the terms obtained from the distributions:
[tex]\[
6x^3 - 21x^2 + 3x - 12x^2 + 42x - 6
\][/tex]
3. Combine like terms:
Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-21x^2 - 12x^2 = -33x^2
\][/tex]
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
3x + 42x = 45x
\][/tex]
So, the polynomial becomes:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
From the choices given, this matches with:
[tex]\[
\boxed{6x^3 - 33x^2 + 45x - 6}
\][/tex]
Therefore, the product of [tex]\((3x - 6)\)[/tex] and [tex]\((2x^2 - 7x + 1)\)[/tex] is [tex]\(\boxed{6x^3 - 33x^2 + 45x - 6}\)[/tex].
