Sure, let's find the product of the given polynomials step-by-step:
We are asked to multiply the polynomials:
[tex]\[
(3x - 6) \quad \text{and} \quad (2x^2 - 7x + 1)
\][/tex]
To find the product, we will distribute each term from the first polynomial with each term from the second polynomial.
1. Distribute [tex]\(3x\)[/tex] to each 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 each 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, we combine all these terms together:
[tex]\[
6x^3 + (-21x^2) + 3x + (-12x^2) + 42x + (-6)
\][/tex]
We collect like terms (combine the coefficients of the same power of [tex]\(x\)[/tex]):
[tex]\[
6x^3 + (-21x^2 - 12x^2) + (3x + 42x) + (-6)
\][/tex]
Simplify the coefficients:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
Therefore, the expanded form of the product is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
So, the correct answer is:
[tex]\[
6x^3 - 33x^2 + 45x - 6
\][/tex]
