1. **State the problem:** Simplify the expression $$\frac{x^2 - 5x + 6}{x^2 - 4x - 5} \times \frac{x^2 - 5}{x^2 + 3x + 2}$$. 2. **Factor all polynomials:** - $x^2 - 5x + 6 = (x - 2)(x - 3)$ - $x^2 - 4x - 5 = (x - 5)(x + 1)$ - $x^2 - 5$ cannot be factored nicely over integers (difference of squares does not apply), so keep as is. - $x^2 + 3x + 2 = (x + 1)(x + 2)$ 3. **Rewrite the expression with factors:** $$\frac{(x - 2)(x - 3)}{(x - 5)(x + 1)} \times \frac{x^2 - 5}{(x + 1)(x + 2)}$$ 4. **Combine into a single fraction:** $$\frac{(x - 2)(x - 3)(x^2 - 5)}{(x - 5)(x + 1)(x + 1)(x + 2)}$$ 5. **Simplify common factors:** There are no common factors to cancel because $x^2 - 5$ does not factor to include $(x + 1)$ or others. 6. **Final simplified expression:** $$\frac{(x - 2)(x - 3)(x^2 - 5)}{(x - 5)(x + 1)^2 (x + 2)}$$ This is the simplified form. **Note:** The domain excludes values that make any denominator zero: $x \neq 5, -1, -2$.