To simplify the expression [tex]\((2x - 3xy)(xy^2 + 2xy - 3x)\)[/tex], we will follow a step-by-step approach.
### Step 1: Expand the expression
First, we expand the product [tex]\((2x - 3xy)(xy^2 + 2xy - 3x)\)[/tex]. This involves distributing each term of the first polynomial to each term of the second polynomial:
[tex]\[
(2x - 3xy)(xy^2 + 2xy - 3x)
\][/tex]
Expanding this:
[tex]\[
2x(xy^2) + 2x(2xy) + 2x(-3x) - 3xy(xy^2) - 3xy(2xy) - 3xy(-3x)
\][/tex]
### Step 2: Simplify each term
Now we simplify each term in the expression:
1. [tex]\(2x \cdot xy^2 = 2x^2y^2\)[/tex]
2. [tex]\(2x \cdot 2xy = 4x^2y\)[/tex]
3. [tex]\(2x \cdot (-3x) = -6x^2\)[/tex]
4. [tex]\(-3xy \cdot xy^2 = -3x^2y^3\)[/tex]
5. [tex]\(-3xy \cdot 2xy = -6x^2y^2\)[/tex]
6. [tex]\(-3xy \cdot (-3x) = 9x^2y\)[/tex]
Substituting these back in, we get:
[tex]\[
2x^2y^2 + 4x^2y - 6x^2 - 3x^2y^3 - 6x^2y^2 + 9x^2y
\][/tex]
### Step 3: Combine like terms
Now we add the like terms together for further simplification:
1. Combine [tex]\(2x^2y^2\)[/tex] and [tex]\(-6x^2y^2\)[/tex]:
[tex]\[
2x^2y^2 - 6x^2y^2 = -4x^2y^2
\][/tex]
2. Combine [tex]\(4x^2y\)[/tex] and [tex]\(9x^2y\)[/tex]:
[tex]\[
4x^2y + 9x^2y = 13x^2y
\][/tex]
3. Combining all, we do not forget other terms:
[tex]\[
-3x^2y^3 + (-4x^2y^2) + 13x^2y - 6x^2
\][/tex]
Putting this all together, our combined simplified expression is:
[tex]\[
-x^2(3y - 2)(y^2 + 2y - 3)
\][/tex]
### Final Answer
The simplified form of the given expression [tex]\((2x - 3xy)(xy^2 + 2xy - 3x)\)[/tex] is:
[tex]\[
-x^2(3y - 2)(y^2 + 2y - 3)
\][/tex]
