1. **State the problem:** Simplify the expression $x^2 (x^{-1} + y^{-1}) \cdot xy (x+y)^2$. 2. **Recall the rules:** - $x^{-1} = \frac{1}{x}$ and $y^{-1} = \frac{1}{y}$. - When multiplying powers with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$. - Distributive property applies for sums. 3. **Rewrite the expression using negative exponents:** $$x^2 \left(\frac{1}{x} + \frac{1}{y}\right) \cdot xy (x+y)^2$$ 4. **Simplify inside the parentheses:** $$\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x+y}{xy}$$ 5. **Substitute back:** $$x^2 \cdot \frac{x+y}{xy} \cdot xy (x+y)^2$$ 6. **Cancel common factors:** $$x^2 \cdot \frac{x+y}{\cancel{xy}} \cdot \cancel{xy} (x+y)^2 = x^2 (x+y) (x+y)^2$$ 7. **Combine like terms:** $$(x+y)(x+y)^2 = (x+y)^3$$ 8. **Final simplified expression:** $$x^2 (x+y)^3$$ **Answer:** $x^2 (x+y)^3$