1. **State the problem:** Simplify the expression $$2x \cdot \frac{(x+y)(x-y)+2y^2}{2y(x-y)} \div \frac{x^2+y^2}{xy-y^2}$$. 2. **Rewrite the division as multiplication by the reciprocal:** $$2x \cdot \frac{(x+y)(x-y)+2y^2}{2y(x-y)} \times \frac{xy - y^2}{x^2 + y^2}$$ 3. **Simplify the numerator of the first fraction:** $$(x+y)(x-y) + 2y^2 = (x^2 - y^2) + 2y^2 = x^2 + y^2$$ 4. **Substitute back:** $$2x \cdot \frac{x^2 + y^2}{2y(x-y)} \times \frac{xy - y^2}{x^2 + y^2}$$ 5. **Cancel common factors $x^2 + y^2$ in numerator and denominator:** $$2x \cdot \frac{\cancel{x^2 + y^2}}{2y(x-y)} \times \frac{xy - y^2}{\cancel{x^2 + y^2}}$$ 6. **Factor $xy - y^2$ as $y(x - y)$:** $$2x \cdot \frac{1}{2y(x-y)} \times y(x - y)$$ 7. **Multiply all terms:** $$2x \times \frac{1}{2y(x-y)} \times y(x - y) = 2x \times \frac{y(x - y)}{2y(x-y)}$$ 8. **Cancel $y$ and $(x - y)$ in numerator and denominator:** $$2x \times \frac{\cancel{y}\cancel{(x - y)}}{2\cancel{y}\cancel{(x - y)}} = 2x \times \frac{1}{2}$$ 9. **Simplify:** $$2x \times \frac{1}{2} = x$$ **Final answer:** $$x$$