Certainly! Let’s simplify the given expression step by step.
We start with the expression:
[tex]\[ x^2 y (x^2 - y) + x y^2 (4 x y - 2 x) \][/tex]
### Step-by-Step Solution:
1. Distribute terms in each part of the expression:
For the first part, [tex]\( x^2 y (x^2 - y) \)[/tex]:
[tex]\[ x^2 y \cdot x^2 - x^2 y \cdot y = x^4 y - x^2 y^2 \][/tex]
For the second part, [tex]\( x y^2 (4 x y - 2 x) \)[/tex]:
[tex]\[ x y^2 \cdot 4 x y - x y^2 \cdot 2 x = 4 x^2 y^3 - 2 x^2 y^2 \][/tex]
2. Combine the results from both parts:
We now put together the results from the previous step:
[tex]\[ x^4 y - x^2 y^2 + 4 x^2 y^3 - 2 x^2 y^2 \][/tex]
3. Combine like terms:
Grouping the [tex]\( y^2 \)[/tex] terms, we get:
[tex]\[ x^4 y + 4 x^2 y^3 - x^2 y^2 - 2 x^2 y^2 \][/tex]
[tex]\[ x^4 y + 4 x^2 y^3 - 3 x^2 y^2 \][/tex]
4. Factor out the common term:
We notice that [tex]\( x^2 y \)[/tex] is a common factor in all terms:
[tex]\[ x^2 y (x^2 + 4 y^2 - 3 y) \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ x^2 y (x^2 + 4 y^2 - 3 y) \][/tex]
That's the final simplified expression for the given problem.
