(13) \( \frac{8 d^{2}+24 d}{20} \div(d+3)^{2} \)

To simplify the expression: \[ \frac{8d^{2} + 24d}{20} \div (d + 3)^{2} \] Follow these steps: 1. **Rewrite the Division as Multiplication:** Dividing by \((d + 3)^2\) is the same as multiplying by \(\frac{1}{(d + 3)^2}\): \[ \frac{8d^{2} + 24d}{20} \times \frac{1}{(d + 3)^2} \] 2. **Factor the Numerator:** Factor out the greatest common factor (GCF) from \(8d^{2} + 24d\): \[ 8d^{2} + 24d = 8d(d + 3) \] So, the expression becomes: \[ \frac{8d(d + 3)}{20} \times \frac{1}{(d + 3)^2} \] 3. **Simplify the Expression:** Combine the fractions: \[ \frac{8d(d + 3)}{20(d + 3)^2} \] Cancel out one \((d + 3)\) term from the numerator and denominator: \[ \frac{8d}{20(d + 3)} \] 4. **Reduce the Fraction:** Simplify \(\frac{8}{20}\) by dividing both numerator and denominator by 4: \[ \frac{2}{5} \] So, the expression simplifies to: \[ \frac{2d}{5(d + 3)} \] **Final Simplified Form:** \[ \boxed{\dfrac{2d}{5(d+3)}} \]