1. **State the problem:** Simplify the expression $$\frac{2}{x-3} + \frac{3}{x+2} - \frac{4x-7}{x^2 - x - 6}$$. 2. **Identify the denominator factorization:** The quadratic in the denominator is $$x^2 - x - 6$$. Factor it: $$x^2 - x - 6 = (x-3)(x+2)$$. 3. **Rewrite the expression with common denominators:** $$\frac{2}{x-3} + \frac{3}{x+2} - \frac{4x-7}{(x-3)(x+2)}$$ 4. **Express all terms with the common denominator $(x-3)(x+2)$:** $$\frac{2(x+2)}{(x-3)(x+2)} + \frac{3(x-3)}{(x-3)(x+2)} - \frac{4x-7}{(x-3)(x+2)}$$ 5. **Combine the numerators:** $$\frac{2(x+2) + 3(x-3) - (4x-7)}{(x-3)(x+2)}$$ 6. **Expand the numerators:** $$2x + 4 + 3x - 9 - 4x + 7$$ 7. **Simplify the numerator:** $$2x + 3x - 4x + 4 - 9 + 7 = (2x + 3x - 4x) + (4 - 9 + 7) = x + 2$$ 8. **Final simplified expression:** $$\frac{x + 2}{(x-3)(x+2)}$$ 9. **Cancel common factors:** The numerator and denominator share a factor $(x+2)$, so cancel it (noting $x \neq -2$): $$\frac{1}{x-3}$$ **Answer:** $$\frac{1}{x-3}$$ This simplification holds for all $x$ except $x=3$ and $x=-2$ where the original expression is undefined.