To solve the expression [tex]\(\frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \div \frac{x^2 y - y^2}{y + 2x}\)[/tex], follow these steps:
### Step 1: Simplify each fraction separately.
1. Simplify [tex]\(\frac{x^2 - 4y^2}{x + y}\)[/tex]
Notice that [tex]\(x^2 - 4y^2\)[/tex] is a difference of squares:
[tex]\[
x^2 - 4y^2 = (x - 2y)(x + 2y)
\][/tex]
Therefore,
[tex]\[
\frac{x^2 - 4y^2}{x + y} = \frac{(x - 2y)(x + 2y)}{x + y}
\][/tex]
This simplifies to:
[tex]\[
x - 2y \quad \text{if} \quad x + y \neq 0
\][/tex]
2. Simplify [tex]\(\frac{x^2 - xy}{2x^2 + 5xy + 2y^2}\)[/tex]
The expression [tex]\(x^2 - xy\)[/tex] can be factored as:
[tex]\[
x^2 - xy = x(x - y)
\][/tex]
We factor the denominator [tex]\(2x^2 + 5xy + 2y^2\)[/tex]. This can be recognized as a quadratic trinomial which factors as:
[tex]\[
2x^2 + 5xy + 2y^2 = (2x + y)(x + 2y)
\][/tex]
Therefore,
[tex]\[
\frac{x^2 - xy}{2x^2 + 5xy + 2y^2} = \frac{x(x - y)}{(2x + y)(x + 2y)}
\][/tex]
3. Simplify [tex]\(\frac{x^2 y - y^2}{y + 2x}\)[/tex]
Factor the numerator [tex]\(x^2 y - y^2\)[/tex]:
[tex]\[
x^2 y - y^2 = y(x^2 - y) = y(x + y)(x - y)
\][/tex]
Thus,
[tex]\[
\frac{x^2 y - y^2}{y + 2x} = \frac{y(x + y)(x - y)}{y + 2x}
\][/tex]
### Step 2: Perform the division and multiplication.
Since division by a fraction is equivalent to multiplication by its reciprocal, we rewrite the operation:
[tex]\[
\frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \div \frac{x^2 y - y^2}{y + 2x} = \frac{x^2 - 4y^2}{x + y} \times \frac{x^2 - xy}{2x^2 + 5xy + 2y^2} \times \frac{y + 2x}{x^2 y - y^2}
\][/tex]
Substitute the simplified forms we obtained:
[tex]\[
(x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)}
\][/tex]
### Step 3: Simplify the resulting expression.
1. Combine similar terms and cancel:
[tex]\[
= (x - 2y) \times \frac{x(x - y)}{(2x + y)(x + 2y)} \times \frac{y + 2x}{y(x - y)}
\][/tex]
2. Notice that [tex]\( (x - y) \)[/tex] appears in both numerator and denominator, so they cancel:
[tex]\[
= (x - 2y) \times \frac{x}{(2x + y)(x + 2y) \times y}
\][/tex]
3. Simplify further, grouping like factors:
[tex]\[
= \frac{x(x - 2y)}{(2x + y)y}
\][/tex]
### Step 4: Write the final simplified expression.
Bringing all these simplifications together, the final expression becomes:
[tex]\[
\frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)}
\][/tex]
Hence the solution to the given mathematical expression is:
[tex]\[
\frac{x(x^2 - 3xy + 2y^2)}{y(x^3 + x^2 y - xy - y^2)}
\][/tex]
This represents the detailed step-by-step simplification process for the given expression.
