To find the product of the expressions [tex]\((3x - 6)\)[/tex] and [tex]\((2x^2 - 7x + 1)\)[/tex], we need to perform polynomial multiplication.
Let's break it down step-by-step:
1. Distribute [tex]\(3x\)[/tex] to each term in the second polynomial:
[tex]\[
3x \cdot (2x^2 - 7x + 1) = 3x \cdot 2x^2 + 3x \cdot (-7x) + 3x \cdot 1
\][/tex]
[tex]\[
= 6x^3 - 21x^2 + 3x
\][/tex]
2. Distribute [tex]\(-6\)[/tex] to each term in the second polynomial:
[tex]\[
-6 \cdot (2x^2 - 7x + 1) = -6 \cdot 2x^2 + (-6) \cdot (-7x) + (-6) \cdot 1
\][/tex]
[tex]\[
= -12x^2 + 42x - 6
\][/tex]
3. Add the results of the two distributions together:
[tex]\[
(6x^3 - 21x^2 + 3x) + (-12x^2 + 42x - 6)
\][/tex]
4. Combine like terms:
- The [tex]\(x^3\)[/tex] term: [tex]\(6x^3\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-21x^2 - 12x^2 = -33x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(3x + 42x = 45x\)[/tex]
- The constant term: [tex]\(-6\)[/tex]
So, the final expanded and simplified expression is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Therefore, the product [tex]\((3x - 6)(2x^2 - 7x + 1)\)[/tex] is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
From the given options, the correct one is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
