2025 AMC 10B Problem 15

The sum

\[\sum_{k=1}^{\infty} \frac{1}{k^3 + 6k^2 + 8k}\]

can be expressed as \dfrac{a}{b}, where a and b are relatively prime positive integers. What is a + b?

\textbf{(A)} 89 \qquad \textbf{(B)} 97 \qquad \textbf{(C)} 102 \qquad \textbf{(D)} 107 \qquad \textbf{(E)} 129

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.