Alright, let's proceed step by step to simplify the given mathematical expression:
[tex]\[ x^2 y\left(x^2 - y\right) + x y^2 (4 x y - 2 x) \][/tex]
1. First Term: Expand [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]
2. Second Term: Expand [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]
3. Combine and Simplify All Terms
[tex]\[
x^4 y - x^2 y^2 + 4 x^2 y^3 - 2 x^2 y^2
\][/tex]
Notice the common factor [tex]\( x^2 y^2 \)[/tex] in the terms [tex]\(- x^2 y^2\)[/tex] and [tex]\(- 2 x^2 y^2\)[/tex]:
[tex]\[
x^4 y - x^2 y^2 - 2 x^2 y^2 + 4 x^2 y^3
\][/tex]
Combine like terms:
[tex]\[
x^4 y + 4x^2 y^3 - 3x^2 y^2
\][/tex]
4. Factor out Common Terms
Factor out the common term [tex]\( x^2 y \)[/tex]:
[tex]\[
x^2 y (x^2 + 4 y^2 - 3y)
\][/tex]
Thus, the simplified form of the given expression:
[tex]\[ x^2 y (x^2 + 4 y^2 - 3 y) \][/tex]
As a result, the simplified expression is:
[tex]\[ x^2 y (x^2 + 4 y^2 - 3 y) \][/tex]
