1. **State the problem:** Simplify the expression $$\frac{3}{x-3} - \frac{5}{x+2}$$. 2. **Find a common denominator:** The denominators are $x-3$ and $x+2$. The common denominator is their product: $$(x-3)(x+2)$$. 3. **Rewrite each fraction with the common denominator:** $$\frac{3}{x-3} = \frac{3(x+2)}{(x-3)(x+2)}$$ $$\frac{5}{x+2} = \frac{5(x-3)}{(x+2)(x-3)}$$ 4. **Subtract the numerators:** $$\frac{3(x+2)}{(x-3)(x+2)} - \frac{5(x-3)}{(x+2)(x-3)} = \frac{3(x+2) - 5(x-3)}{(x-3)(x+2)}$$ 5. **Expand the numerators:** $$3(x+2) = 3x + 6$$ $$5(x-3) = 5x - 15$$ 6. **Substitute back and simplify numerator:** $$3x + 6 - (5x - 15) = 3x + 6 - 5x + 15 = (3x - 5x) + (6 + 15) = -2x + 21$$ 7. **Final simplified expression:** $$\frac{-2x + 21}{(x-3)(x+2)}$$ 8. **Optional:** Factor out $-1$ from numerator for a cleaner look: $$\frac{-1(2x - 21)}{(x-3)(x+2)}$$ **Answer:** $$\boxed{\frac{-2x + 21}{(x-3)(x+2)}}$$