1. **State the problem:** Simplify the function $f(x) = \frac{x^{3} + 2x^{2}}{x^{2} - 1}$. 2. **Recall the formula and rules:** To simplify a rational function, factor numerator and denominator and cancel common factors. 3. **Factor numerator:** $$x^{3} + 2x^{2} = x^{2}(x + 2)$$ 4. **Factor denominator:** $$x^{2} - 1 = (x - 1)(x + 1)$$ 5. **Rewrite the function:** $$f(x) = \frac{x^{2}(x + 2)}{(x - 1)(x + 1)}$$ 6. **Check for common factors:** None of $x^{2}$, $x + 2$ match $x - 1$ or $x + 1$, so no cancellation. 7. **Final simplified form:** $$f(x) = \frac{x^{2}(x + 2)}{(x - 1)(x + 1)}$$ **Note:** The domain excludes $x = 1$ and $x = -1$ because denominator is zero there.