Certainly! Let's go through the step-by-step process of expanding the expression:
We need to find the product of:
[tex]\[
(3x - 6) \left(2x^2 - 7x + 1\right)
\][/tex]
First, we distribute each term in the first polynomial to each term in the second polynomial.
1. Distribute \(3x\) to each term in \(2x^2 - 7x + 1\):
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
[tex]\[
3x \cdot (-7x) = -21x^2
\][/tex]
[tex]\[
3x \cdot 1 = 3x
\][/tex]
2. Distribute \(-6\) to each term in \(2x^2 - 7x + 1\):
[tex]\[
-6 \cdot 2x^2 = -12x^2
\][/tex]
[tex]\[
-6 \cdot (-7x) = 42x
\][/tex]
[tex]\[
-6 \cdot 1 = -6
\][/tex]
Now, combine all these products:
[tex]\[
6x^3 - 21x^2 + 3x - 12x^2 + 42x - 6
\][/tex]
Next, combine like terms:
[tex]\[
6x^3 + (-21x^2 - 12x^2) + (3x + 42x) - 6
\][/tex]
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
So, the expanded form of the product of \((3x - 6)\left(2x^2 - 7x + 1\right)\) is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Therefore, the correct answer is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
