To expand the expression [tex]\((-x y + 2 x - 2)(3 x y + 3 x + z)\)[/tex], we need to apply the distributive property (also known as the FOIL method in the context of binomials), which involves multiplying each term in the first polynomial by each term in the second polynomial and then simplifying. Here's the step-by-step process:
1. Distribute the terms:
[tex]\[
(-x y + 2 x - 2)(3 x y + 3 x + z)
\][/tex]
This means we will distribute each term in the first polynomial across each term in the second polynomial:
2. Multiply each pair of terms:
- [tex]\((-x y) \cdot (3 x y) = -3 x^2 y^2\)[/tex]
- [tex]\((-x y) \cdot (3 x) = -3 x^2 y\)[/tex]
- [tex]\((-x y) \cdot (z) = -x y z\)[/tex]
- [tex]\((2 x) \cdot (3 x y) = 6 x^2 y\)[/tex]
- [tex]\((2 x) \cdot (3 x) = 6 x^2\)[/tex]
- [tex]\((2 x) \cdot (z) = 2 x z\)[/tex]
- [tex]\((-2) \cdot (3 x y) = -6 x y\)[/tex]
- [tex]\((-2) \cdot (3 x) = -6 x\)[/tex]
- [tex]\((-2) \cdot (z) = -2 z\)[/tex]
3. Combine all the products:
[tex]\[
-3 x^2 y^2 - 3 x^2 y - x y z + 6 x^2 y + 6 x^2 + 2 x z - 6 x y - 6 x - 2 z
\][/tex]
4. Combine like terms:
Let's group the like terms together and then combine them:
[tex]\[
(-3 x^2 y^2) + (-3 x^2 y + 6 x^2 y) + (6 x^2) + (-x y z) + (2 x z) + (-6 x y) + (-6 x) + (-2 z)
\][/tex]
Here, we can combine:
- [tex]\(-3 x^2 y + 6 x^2 y = 3 x^2 y\)[/tex]
The remaining terms do not have like terms to combine with, so we write them as is.
5. Final simplified expression:
[tex]\[
-3 x^2 y^2 + 3 x^2 y + 6 x^2 - x y z - 6 x y + 2 x z - 6 x - 2 z
\][/tex]
Thus, the expanded form of [tex]\((-x y + 2 x - 2)(3 x y + 3 x + z)\)[/tex] is:
[tex]\[
-3 x^2 y^2 + 3 x^2 y + 6 x^2 - x y z - 6 x y + 2 x z - 6 x - 2 z
\][/tex]
