1. **Stating the problem:** Simplify the expression $$\frac{x^2 - 5x + 6}{x^2 + x - 12} - 5$$. 2. **Factor the polynomials:** - Numerator of the fraction: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ because $-2 \times -3 = 6$ and $-2 + -3 = -5$. - Denominator of the fraction: $$x^2 + x - 12 = (x + 4)(x - 3)$$ because $4 \times -3 = -12$ and $4 + (-3) = 1$. 3. **Rewrite the expression with factored forms:** $$\frac{(x - 2)(x - 3)}{(x + 4)(x - 3)} - 5$$ 4. **Cancel common factors:** The factor $(x - 3)$ appears in numerator and denominator, so we cancel it: $$\frac{\cancel{(x - 3)}(x - 2)}{(x + 4)\cancel{(x - 3)}} - 5 = \frac{x - 2}{x + 4} - 5$$ 5. **Rewrite 5 as a fraction with denominator $(x + 4)$:** $$5 = \frac{5(x + 4)}{x + 4} = \frac{5x + 20}{x + 4}$$ 6. **Combine the fractions:** $$\frac{x - 2}{x + 4} - \frac{5x + 20}{x + 4} = \frac{x - 2 - (5x + 20)}{x + 4}$$ 7. **Simplify the numerator:** $$x - 2 - 5x - 20 = (x - 5x) + (-2 - 20) = -4x - 22$$ 8. **Final simplified expression:** $$\frac{-4x - 22}{x + 4}$$ 9. **Factor numerator if possible:** $$-4x - 22 = -2(2x + 11)$$ So the expression can be written as: $$\frac{-2(2x + 11)}{x + 4}$$ **Answer:** $$\boxed{\frac{-2(2x + 11)}{x + 4}}$$