Divide. Enter your answer as a single completely simplified rational expression with nonnegative exponents. \[ \frac{n^{2}+12 n+32}{n^{2}+4 n} \div\left(4 n^{2}+16 n\right) \]

To solve the expression \(\frac{n^{2}+12 n+32}{n^{2}+4 n} \div\left(4 n^{2}+16 n\right)\), we first rewrite the division as multiplication by the reciprocal: \[ \frac{n^{2}+12 n+32}{n^{2}+4 n} \cdot \frac{1}{4 n^{2}+16 n} \] Next, let’s factor each part of the expression. 1. **Factor \(n^{2}+12n+32\)**: \[ n^{2}+12n+32 = (n+4)(n+8) \] 2. **Factor \(n^{2}+4n\)**: \[ n^{2}+4n = n(n+4) \] 3. **Factor \(4n^{2}+16n\)**: \[ 4n^{2}+16n = 4n(n+4) \] Now we can rewrite the original expression using these factors: \[ \frac{(n+4)(n+8)}{n(n+4)} \cdot \frac{1}{4n(n+4)} \] Next, simplify by cancelling the common factors: - The \(n+4\) in the numerator and denominator cancels out. Now we have: \[ \frac{(n+8)}{n} \cdot \frac{1}{4n} = \frac{n+8}{n \cdot 4n} = \frac{n+8}{4n^2} \] So, the final simplified expression is: \[ \frac{n+8}{4n^2} \] This is a completely simplified rational expression with nonnegative exponents.