2017 AMC 10A Problem 24

Because the three distinct roots of the cubic $g$ are also roots of the monic quartic $f,$ we can write $$f(x)=g(x)(x-r)$$ for some real number $r.$ The coefficient of $x$ in this product is $10-r,$ so $10-r=100$ and $r=-90.$ The coefficient of $x^3$ is $a-r,$ so $a-r=1$ and $a=-89.$ Therefore, $$\begin{aligned} f(1)&=g(1)(1-r)\\ &=(1-89+1+10)(91)\\ &=(-77)(91)\\ &=-7007. \end{aligned}$$ Thus, C is the correct answer.