Divide. Enter your answer as a single completely simplified rational expression with nonnegative exponents. \[ \frac{5 x^{2}}{x^{2}-15 x} \div \frac{15 x^{2}+15 x}{x^{2}-4 x-5} \]

To simplify the expression \(\frac{5 x^{2}}{x^{2}-15 x} \div \frac{15 x^{2}+15 x}{x^{2}-4 x-5}\), we start by rewriting the division as multiplication by the reciprocal: \[ \frac{5 x^{2}}{x^{2}-15 x} \cdot \frac{x^{2}-4 x-5}{15 x^{2}+15 x} \] Now we factor each part: 1. **For \(x^{2}-15x\)**, we factor out \(x\): \[ x^{2}-15x = x(x-15) \] 2. **For \(x^{2}-4x-5\)**, we find the factors: \[ x^{2}-4x-5 = (x-5)(x+1) \] 3. **For \(15x^{2}+15x\)**, we can factor out \(15x\): \[ 15x^{2}+15x = 15x(x+1) \] Putting these together in our expression, we get: \[ \frac{5x^{2}}{x(x-15)} \cdot \frac{(x-5)(x+1)}{15x(x+1)} \] Next, we multiply the fractions: \[ \frac{5x^{2} \cdot (x-5)(x+1)}{x(x-15) \cdot 15x(x+1)} \] Now we can cancel out common factors. Notice that \(x\) in the numerator and denominator cancels out, as does \(x+1\): \[ \frac{5(x-5)}{15(x-15)} \] Next, we simplify the coefficients: \[ \frac{5}{15} = \frac{1}{3} \] Thus, the expression simplifies to: \[ \frac{x-5}{3(x-15)} \] To summarize the final answer, the completely simplified rational expression with nonnegative exponents is: \[ \frac{x-5}{3(x-15)} \]