Simplify. \[ \frac{\frac{x^{2}-x-2}{x^{2}-2 x-3}}{\frac{x^{2}+10 x+24}{x^{2}-4 x-32}} \] \( \frac{\frac{x^{2}-x-2}{x^{2}-2 x-3}}{\frac{x^{2}+10 x+24}{x^{2}-4 x-32}}=\square \) (Simplify your answer)

To simplify the expression \[ \frac{\frac{x^{2}-x-2}{x^{2}-2 x-3}}{\frac{x^{2}+10 x+24}{x^{2}-4 x-32}}, \] we start by factoring each polynomial. 1. **Factoring the numerator and denominator:** - For \( x^{2} - x - 2 \): \[ x^{2} - x - 2 = (x - 2)(x + 1). \] - For \( x^{2} - 2x - 3 \): \[ x^{2} - 2x - 3 = (x - 3)(x + 1). \] - For \( x^{2} + 10x + 24 \): \[ x^{2} + 10x + 24 = (x + 6)(x + 4). \] - For \( x^{2} - 4x - 32 \): \[ x^{2} - 4x - 32 = (x - 8)(x + 4). \] 2. **Substituting the factored forms into the expression:** \[ \frac{\frac{(x - 2)(x + 1)}{(x - 3)(x + 1)}}{\frac{(x + 6)(x + 4)}{(x - 8)(x + 4)}}. \] 3. **Dividing by the fraction in the denominator:** This is equivalent to multiplying by the reciprocal: \[ = \frac{(x - 2)(x + 1)}{(x - 3)(x + 1)} \times \frac{(x - 8)(x + 4)}{(x + 6)(x + 4)}. \] 4. **Now, we can simplify:** - Cancel \( (x + 1) \) from the numerator and denominator: \[ = \frac{(x - 2)}{(x - 3)} \times \frac{(x - 8)}{(x + 6)}. \] - Cancel \( (x + 4) \) from the numerator and denominator: \[ = \frac{(x - 2)(x - 8)}{(x - 3)(x + 6)}. \] 5. **Final Result:** The expression simplifies to: \[ \frac{(x - 2)(x - 8)}{(x - 3)(x + 6)}. \] Thus, the answer is \[ \frac{(x - 2)(x - 8)}{(x - 3)(x + 6)}. \]