To find the product of the given expressions:
[tex]\[ (3x - 6)\left(2x^2 - 7x + 1\right) \][/tex]
we need to multiply each term in the first polynomial by each term in the second polynomial, and then combine like terms.
### Step-by-step solution:
1. Distribute [tex]\(3x\)[/tex] to each term in the second polynomial:
[tex]\[
3x \cdot 2x^2 = 6x^3
\][/tex]
[tex]\[
3x \cdot (-7x) = -21x^2
\][/tex]
[tex]\[
3x \cdot 1 = 3x
\][/tex]
So, the result from [tex]\(3x\)[/tex] distribution is:
[tex]\[
6x^3 - 21x^2 + 3x
\][/tex]
2. Distribute [tex]\(-6\)[/tex] to each term in the second polynomial:
[tex]\[
-6 \cdot 2x^2 = -12x^2
\][/tex]
[tex]\[
-6 \cdot (-7x) = 42x
\][/tex]
[tex]\[
-6 \cdot 1 = -6
\][/tex]
So, the result from [tex]\(-6\)[/tex] distribution is:
[tex]\[
-12x^2 + 42x - 6
\][/tex]
3. Combine all the results from steps 1 and 2:
[tex]\[
6x^3 - 21x^2 + 3x - 12x^2 + 42x - 6
\][/tex]
4. Combine like terms:
[tex]\[
6x^3 + (-21x^2 - 12x^2) + (3x + 42x) - 6
\][/tex]
Simplifying the like terms:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Therefore, the product of the given expressions is:
[tex]\[ 6x^3 - 33x^2 + 45x - 6 \][/tex]
We can now identify the correct option from the multiple choices:
[tex]\[ \boxed{6x^3 - 33x^2 + 45x - 6} \][/tex]
