1. **State the problem:** Simplify the expression $$\frac{2}{x^2 - 2x} - \frac{5}{x^2 + x - 6}$$. 2. **Factor the denominators:** - For $$x^2 - 2x$$, factor out $$x$$: $$x^2 - 2x = x(x - 2)$$. - For $$x^2 + x - 6$$, find factors of $$-6$$ that add to $$1$$: $$x^2 + x - 6 = (x + 3)(x - 2)$$. 3. **Rewrite the expression with factored denominators:** $$\frac{2}{x(x - 2)} - \frac{5}{(x + 3)(x - 2)}$$. 4. **Find the common denominator:** The least common denominator (LCD) is $$x(x - 2)(x + 3)$$. 5. **Rewrite each fraction with the LCD:** $$\frac{2}{x(x - 2)} = \frac{2(x + 3)}{x(x - 2)(x + 3)}$$ $$\frac{5}{(x + 3)(x - 2)} = \frac{5x}{x(x - 2)(x + 3)}$$ 6. **Combine the fractions:** $$\frac{2(x + 3)}{x(x - 2)(x + 3)} - \frac{5x}{x(x - 2)(x + 3)} = \frac{2(x + 3) - 5x}{x(x - 2)(x + 3)}$$ 7. **Simplify the numerator:** $$2(x + 3) - 5x = 2x + 6 - 5x = -3x + 6$$ 8. **Factor the numerator:** $$-3x + 6 = -3(x - 2)$$ 9. **Substitute back:** $$\frac{-3(x - 2)}{x(x - 2)(x + 3)}$$ 10. **Cancel the common factor $$(x - 2)$$:** $$\frac{-3\cancel{(x - 2)}}{x\cancel{(x - 2)}(x + 3)} = \frac{-3}{x(x + 3)}$$ **Final answer:** $$\boxed{\frac{-3}{x(x + 3)}}$$