AMC 12 2017a Problem 23

For certain real numbers $a$, $b$, and $c$, the polynomial $$g(x) = x^3 + ax^2 + x + 10$$has three distinct roots, and each root of $g(x)$ is also a root of the polynomial $$f(x) = x^4 + x^3 + bx^2 + 100x + c.$$What is $f(1)$?

\textbf{(A)}\ -9009 \textbf{(B)}\ -8008 \textbf{(C)}\ -7007 \textbf{(D)}\ -6006 \textbf{(E)}\ -5005