1. **State the problem:** Simplify the expression $$\frac{x^2 + x - 6}{x^2 - x - 12} \times \frac{x^2 + x - 20}{x^2 - 4x + 4}$$. 2. **Factor each polynomial:** - Numerator 1: $x^2 + x - 6 = (x + 3)(x - 2)$ - Denominator 1: $x^2 - x - 12 = (x - 4)(x + 3)$ - Numerator 2: $x^2 + x - 20 = (x + 5)(x - 4)$ - Denominator 2: $x^2 - 4x + 4 = (x - 2)^2$ 3. **Rewrite the expression with factors:** $$\frac{(x + 3)(x - 2)}{(x - 4)(x + 3)} \times \frac{(x + 5)(x - 4)}{(x - 2)^2}$$ 4. **Multiply the fractions:** $$\frac{(x + 3)(x - 2)(x + 5)(x - 4)}{(x - 4)(x + 3)(x - 2)^2}$$ 5. **Cancel common factors:** - Cancel $(x + 3)$ numerator and denominator - Cancel $(x - 4)$ numerator and denominator - Cancel one $(x - 2)$ from numerator and denominator Intermediate step showing cancellation: $$\frac{\cancel{(x + 3)}\cancel{(x - 2)}(x + 5)\cancel{(x - 4)}}{\cancel{(x - 4)}\cancel{(x + 3)}(x - 2)\cancel{(x - 2)}} = \frac{(x + 5)}{(x - 2)}$$ 6. **Final simplified expression:** $$\boxed{\frac{x + 5}{x - 2}}$$ This is the simplified form of the original expression, valid for all $x$ except where the original denominators are zero (i.e., $x \neq 4, -3, 2$).