To expand the given expression [tex]\((3x + 2)(-x^2 - 3x + 1)\)[/tex], we need to perform polynomial multiplication. We will use the distributive property, also known as the distributive law of multiplication, to expand this expression thoroughly. Here are the steps:
1. Distribute each term in the first polynomial [tex]\(3x + 2\)[/tex] to each term in the second polynomial [tex]\(-x^2 - 3x + 1\)[/tex]:
[tex]\[
(3x + 2)(-x^2 - 3x + 1) = 3x(-x^2 - 3x + 1) + 2(-x^2 - 3x + 1)
\][/tex]
2. Distribute [tex]\(3x\)[/tex] to each term in [tex]\(-x^2 - 3x + 1\)[/tex]:
[tex]\[
3x(-x^2) + 3x(-3x) + 3x(1)
\][/tex]
[tex]\[
= -3x^3 - 9x^2 + 3x
\][/tex]
3. Distribute [tex]\(2\)[/tex] to each term in [tex]\(-x^2 - 3x + 1\)[/tex]:
[tex]\[
2(-x^2) + 2(-3x) + 2(1)
\][/tex]
[tex]\[
= -2x^2 - 6x + 2
\][/tex]
4. Combine the results from these distributions:
[tex]\[
-3x^3 - 9x^2 + 3x + (-2x^2 - 6x + 2)
\][/tex]
5. Combine like terms:
[tex]\[
-3x^3 + (-9x^2 - 2x^2) + (3x - 6x) + 2
\][/tex]
[tex]\[
= -3x^3 - 11x^2 - 3x + 2
\][/tex]
So, the expanded form of the expression [tex]\((3x + 2)(-x^2 - 3x + 1)\)[/tex] is:
[tex]\[
-3x^3 - 11x^2 - 3x + 2
\][/tex]
