Simplify: (x2 - y2) / (x+y) ÷ (2y-2x) / (3xy)

The numerator $x^2 - y^2$ is a difference of squares, which factors to $(x-y)(x+y)$.
The denominator $x^2 + xy$ has a common factor of $x$, which factors to $x(x+y)$.
The numerator $2y - 2x$ has a common factor of 2, which factors to $2(y-x)$. We can also write $y-x$ as $-(x-y)$.
The denominator $xy$ is already in its simplest form.

Substitute the factored forms into the expression:

$$\frac{(x-y)(x+y)}{x(x+y)} \times \frac{xy}{2(-(x-y))}$$ $$\frac{(x-y)(x+y)}{x(x+y)} \times \frac{xy}{-2(x-y)}$$

We can cancel $(x+y)$ from the first fraction's numerator and denominator.
We can cancel $x$ from the denominator of the first fraction and the numerator of the second fraction.
We can cancel $(x-y)$ from the numerator of the first fraction and the denominator of the second fraction.

$$\frac{\cancel{(x-y)}\cancel{(x+y)}}{\cancel{x}\cancel{(x+y)}} \times \frac{\cancel{x}y}{-2\cancel{(x-y)}}$$

$$1 \times \frac{y}{-2} = -\frac{y}{2}$$

The simplified expression is:

$$-\frac{y}{2}$$