1. **State the problem:** Simplify the expression $$\frac{2}{x-3} + \frac{3}{x+2}$$. 2. **Formula and rules:** To add rational expressions, find a common denominator, which is the least common multiple (LCM) of the denominators. 3. **Find the common denominator:** The denominators are $x-3$ and $x+2$, so the common denominator is $(x-3)(x+2)$. 4. **Rewrite each fraction with the common denominator:** $$\frac{2}{x-3} = \frac{2(x+2)}{(x-3)(x+2)}$$ $$\frac{3}{x+2} = \frac{3(x-3)}{(x+2)(x-3)}$$ 5. **Add the numerators:** $$\frac{2(x+2) + 3(x-3)}{(x-3)(x+2)}$$ 6. **Simplify the numerator:** $$2(x+2) + 3(x-3) = 2x + 4 + 3x - 9 = 5x - 5$$ 7. **Factor the numerator:** $$5x - 5 = 5(x - 1)$$ 8. **Final simplified expression:** $$\frac{5(x-1)}{(x-3)(x+2)}$$ This is the simplified form of the original expression.