AMC 12 2025b Problem 8

There are integers $a$ and $b$ such that the polynomial $x^3 - 5x^2 + ax + b$ has $4+\sqrt{5}$ as a root. What is $a+b$?

$\textbf{(A) } 13 \textbf{(B) } 17 \textbf{(C) } 20 \textbf{(D) } 30 \textbf{(E) } 68$