The problem is to simplify the following expression: $\frac{10xy(x-2y)}{-6x^3y^2(x+y)} \times \frac{30x^4y(y+x)}{5xy^3(x-2y)}$

1. Problem Description

The problem is to simplify the following expression:

$\frac{10xy(x-2y)}{-6x^3y^2(x+y)} \times \frac{30x^4y(y+x)}{5xy^3(x-2y)}$

2. Solution Steps

First, rewrite the expression as a single fraction:

$\frac{10xy(x-2y) \times 30x^4y(y+x)}{-6x^3y^2(x+y) \times 5xy^3(x-2y)}$

Next, multiply the numerators and the denominators:

$\frac{300x^5y^2(x-2y)(y+x)}{-30x^4y^5(x+y)(x-2y)}$

Now, we can simplify by canceling common factors.

First, cancel the common factor $(x-2y)$ from the numerator and the denominator. Note that it is assumed that $x-2y \neq 0$:

$\frac{300x^5y^2(y+x)}{-30x^4y^5(x+y)}$

Next, cancel the common factor $(x+y)$ from the numerator and the denominator. Note that it is assumed that $x+y \neq 0$:

$\frac{300x^5y^2}{-30x^4y^5}$

Next, simplify the numerical coefficients: $\frac{300}{-30} = -10$

$\frac{-10x^5y^2}{x^4y^5}$

Now, simplify the powers of $x$ and $y$:

$\frac{x^5}{x^4} = x^{5-4} = x$

$\frac{y^2}{y^5} = y^{2-5} = y^{-3} = \frac{1}{y^3}$

Therefore, the expression simplifies to:

$-10 \cdot \frac{x}{y^3} = \frac{-10x}{y^3}$

3. Final Answer

The simplified expression is $\frac{-10x}{y^3}$.