24. C.

Let $r \in \mathbf{C}$ be the fourth root of the polynomial $f(x)$. Then
$$
\begin{array}{l}
f(x)=g(x)(x-r) \\
=\left(x^{3}+a x^{2}+x+10\right)(x-r) \\
=x^{4}+(a-r) x^{3}+(1-a r) x^{2}+(10-r) x-10 r \\
=x^{4}+x^{3}+b x^{2}+100 x+c \\
\Rightarrow\left\{\begin{array} { l } 
{ a - r = 1 , } \\
{ 1 - a r = b , } \\
{ 1 0 - r = 1 0 0 , } \\
{ - 1 0 r = c }
\end{array} \Rightarrow \left\{\begin{array}{l}
a=-89, \\
b=-8009, \\
r=-90, \\
c=900
\end{array}\right.\right. \\
\Rightarrow f(x)=x^{4}+x^{3}-8009 x^{2}+100 x+900 \\
\Rightarrow f(1)=-7007 .
\end{array}
$$
24. Given real numbers $a$, $b$, $c$, the polynomial
$$
g(x)=x^{3}+a x^{2}+x+10
$$

has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x)=x^{4}+x^{3}+b x^{2}+100 x+c
$$

What is the value of $f(1)$? 
(A) -9009
(B) -8008
(C) -7007
(D) -6006
(E) -5005