AMC 12 2019b Problem 4

A positive integer $n$ satisfies the equation $(n+1)!+(n+2)!=440\cdot n!$. What is the sum of the digits of $n$?

$\textbf{(A) } 2 \textbf{(B) } 5 \textbf{(C) } 10 \textbf{(D) } 12 \textbf{(E) } 15$