Divide and simplify. \( \frac{x^{2}-4 x-5}{2 x^{2}-50} \div \frac{x^{2}+6 x+5}{x^{2}+10 x+25} \)

To divide and simplify the expression \[ \frac{x^{2}-4 x-5}{2 x^{2}-50} \div \frac{x^{2}+6 x+5}{x^{2}+10 x+25}, \] we first rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2}-4 x-5}{2 x^{2}-50} \cdot \frac{x^{2}+10 x+25}{x^{2}+6 x+5}. \] Next, we can factor the expressions: 1. For \(x^2 - 4x - 5\): \[ x^2 - 4x - 5 = (x - 5)(x + 1). \] 2. For \(2x^2 - 50\): \[ 2x^2 - 50 = 2(x^2 - 25) = 2(x - 5)(x + 5). \] 3. For \(x^2 + 6x + 5\): \[ x^2 + 6x + 5 = (x + 5)(x + 1). \] 4. For \(x^2 + 10x + 25\): \[ x^2 + 10x + 25 = (x + 5)^2. \] Now substituting back into the expression gives us: \[ \frac{(x - 5)(x + 1)}{2(x - 5)(x + 5)} \cdot \frac{(x + 5)^2}{(x + 5)(x + 1)}. \] Now we can simplify step by step: - The \((x + 1)\) in the numerator and denominator cancels. - The \((x - 5)\) in the numerator and denominator cancels. - The \((x + 5)\) in the denominator cancels with one of the \((x + 5)\) factors in the numerator. After canceling, we are left with: \[ \frac{(x + 5)}{2}. \] Thus, the simplified expression is: \[ \frac{x + 5}{2}. \]