Divide. Write your answer in simplest form. \( \frac{4}{d^{2}-3 d} \div\left(d^{2}+9 d+8\right) \)

To solve the division of fractions, we can rewrite the expression as a multiplication by the reciprocal. Therefore, \[ \frac{4}{d^{2}-3d} \div \left(d^{2}+9d+8\right) = \frac{4}{d^{2}-3d} \times \frac{1}{d^{2}+9d+8} \] Next, we need to factor both denominators. 1. **Factoring \(d^{2}-3d\):** \[ d^{2}-3d = d(d-3) \] 2. **Factoring \(d^{2}+9d+8\):** To factor \(d^{2}+9d+8\), we look for two numbers that multiply to 8 (the constant term) and add up to 9 (the linear coefficient). The numbers 8 and 1 fit perfectly. \[ d^{2}+9d+8 = (d+8)(d+1) \] Now substituting the factored forms back into the expression gives us: \[ \frac{4}{d(d-3)} \times \frac{1}{(d+8)(d+1)} \] Now we can multiply the fractions: \[ \frac{4 \cdot 1}{d(d-3)(d+8)(d+1)} \] This simplifies to: \[ \frac{4}{d(d-3)(d+8)(d+1)} \] Finally, since there are no common factors that can be canceled, the answer in simplest form is: \[ \frac{4}{d(d-3)(d+8)(d+1)} \]