1. **State the problem:** Find the middle zero of the function $f(x) = x^5 (x - 1)^2 (x + 8)$ and its multiplicity. 2. **Identify zeros:** The zeros are $x = -8$, $x = 0$, and $x = 1$ with multiplicities 1, 5, and 2 respectively. 3. **Order the zeros:** The zeros in ascending order are $-8$, $0$, and $1$. 4. **Find the middle zero:** The middle zero is $0$. 5. **Multiplicity of middle zero:** From the factor $x^5$, the multiplicity is 5. **Final answer:** The middle zero is $0$ with multiplicity 5.