Divide. \[ \left(32 y z^{2}-8 y^{7} z^{7}\right) \div\left(4 y^{5} z^{3}\right) \]

To divide the expression \( \left(32 y z^{2}-8 y^{7} z^{7}\right) \div\left(4 y^{5} z^{3}\right) \), we can begin by splitting it into two separate fractions: \[ \frac{32 y z^{2}}{4 y^{5} z^{3}} - \frac{8 y^{7} z^{7}}{4 y^{5} z^{3}} \] Now, we can simplify each fraction separately. For the first term: \[ \frac{32 y z^{2}}{4 y^{5} z^{3}} = \frac{32}{4} \cdot \frac{y}{y^{5}} \cdot \frac{z^{2}}{z^{3}} = 8 \cdot y^{1-5} \cdot z^{2-3} = 8 \cdot y^{-4} \cdot z^{-1} = \frac{8}{y^{4} z} \] For the second term: \[ \frac{8 y^{7} z^{7}}{4 y^{5} z^{3}} = \frac{8}{4} \cdot \frac{y^{7}}{y^{5}} \cdot \frac{z^{7}}{z^{3}} = 2 \cdot y^{7-5} \cdot z^{7-3} = 2 \cdot y^{2} \cdot z^{4} = 2y^{2}z^{4} \] Putting it all together: \[ \frac{8}{y^{4} z} - 2y^{2} z^{4} \] Thus, the final answer for the division is: \[ \frac{8}{y^{4} z} - 2y^{2} z^{4} \]