Simplify the equation by canceling common factors: \( \frac{x^{2}-x-6}{x^{2}-6 x-16} \)

To simplify the given rational expression by canceling common factors: \[ \frac{x^{2} - x - 6}{x^{2} - 6x - 16} \] **Step 1: Factor the Numerator** Factor \( x^{2} - x - 6 \). We look for two numbers that multiply to \(-6\) and add up to \(-1\). These numbers are \(-3\) and \(2\). \[ x^{2} - x - 6 = (x - 3)(x + 2) \] **Step 2: Factor the Denominator** Factor \( x^{2} - 6x - 16 \). We look for two numbers that multiply to \(-16\) and add up to \(-6\). These numbers are \(-8\) and \(2\). \[ x^{2} - 6x - 16 = (x - 8)(x + 2) \] **Step 3: Write the Factored Form** Substitute the factored forms back into the original expression: \[ \frac{(x - 3)(x + 2)}{(x - 8)(x + 2)} \] **Step 4: Cancel Common Factors** Notice that \((x + 2)\) is a common factor in both the numerator and the denominator. Assuming \(x \neq -2\) (to avoid division by zero), we can cancel this factor: \[ \frac{\cancel{(x - 3)}\cancel{(x + 2)}}{\cancel{(x + 2)}(x - 8)} = \frac{x - 3}{x - 8} \] **Final Simplified Expression:** \[ \boxed{\dfrac{x - 3}{x - 8}} \]