Sure, let's break down and simplify the given expression step by step.
The expression we need to simplify is:
[tex]\[ x^2 y \left(x^2 - y\right) + x y^2 (4 x y - 2 x) \][/tex]
### Step 1: Expand each part separately
First part:
[tex]\[ x^2 y (x^2 - y) \][/tex]
Distribute [tex]\( x^2 y \)[/tex]:
[tex]\[ x^2 y \cdot x^2 - x^2 y \cdot y \][/tex]
[tex]\[ x^4 y - x^2 y^2 \][/tex]
Second part:
[tex]\[ x y^2 (4 x y - 2 x) \][/tex]
Distribute [tex]\( x y^2 \)[/tex]:
[tex]\[ x y^2 \cdot 4 x y - x y^2 \cdot 2 x \][/tex]
[tex]\[ 4 x^2 y^3 - 2 x^2 y^2 \][/tex]
### Step 2: Combine the expanded parts
Now we combine the results from each part:
[tex]\[ x^4 y - x^2 y^2 + 4 x^2 y^3 - 2 x^2 y^2 \][/tex]
### Step 3: Simplify the combined expression
Combine like terms:
[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]
### Final Expression
So, the simplified form of the given expression is:
[tex]\[ x^4 y + 4 x^2 y^3 - 3 x^2 y^2 \][/tex]
This step-by-step expansion and simplification process offer a clear method to arrive at the expression:
[tex]\[ x^2 y \left(x^2 - y\right) + x y^2(4 x y - 2 x) \][/tex]
