Simplify the rational expression by canceling common factors: \( \begin{array}{l}\frac{x^{2}+7 x+6}{x^{2}-3 x-4} \\ \frac{x+6}{x-4} \\ \frac{x+1}{x-4} \\ \frac{x-4}{x+6}\end{array} \)

To simplify the rational expression by canceling common factors, follow these steps: **Given Expression:** \[ \frac{x^{2} + 7x + 6}{x^{2} - 3x - 4} \] **Step 1: Factor the Numerator and Denominator** 1. **Factor the numerator \(x^{2} + 7x + 6\):** \[ x^{2} + 7x + 6 = (x + 1)(x + 6) \] 2. **Factor the denominator \(x^{2} - 3x - 4\):** \[ x^{2} - 3x - 4 = (x + 1)(x - 4) \] **Step 2: Rewrite the Expression with Factored Forms** \[ \frac{(x + 1)(x + 6)}{(x + 1)(x - 4)} \] **Step 3: Cancel the Common Factor \((x + 1)\)** \[ \frac{\cancel{(x + 1)}(x + 6)}{\cancel{(x + 1)}(x - 4)} = \frac{x + 6}{x - 4} \] **Final Simplified Expression:** \[ \frac{x + 6}{x - 4} \] **Note:** - The original expression is defined for all real numbers except where the denominator is zero. - After simplification, the restrictions are \(x \neq -1\) and \(x \neq 4\).