1. **State the problem:** Simplify the expression $$\frac{2x \cdot x^2}{(x - y)^2}$$. 2. **Write the formula and rules:** When multiplying powers with the same base, add the exponents: $$a^m \cdot a^n = a^{m+n}$$. 3. **Simplify the numerator:** $$2x \cdot x^2 = 2x^{1+2} = 2x^3$$. 4. **Rewrite the expression:** $$\frac{2x^3}{(x - y)^2}$$. 5. **Final simplified form:** The expression cannot be simplified further without additional information, so the simplified expression is: $$\boxed{\frac{2x^3}{(x - y)^2}}$$.