On the graph of $y=(x+2)^4-100$, how many points are there whose coordinates are both negative integers?
A point $(x,y)$ is on the graph if and only if $y=(x+2)^4-100$, so we seek to determine all pairs of negative integers $(x,y)$ which satisfy this equation. We can obtain pairs by plugging in $-1,-2,-3,$ and so on for $x$: \begin{align*}
x=-1 &\Rightarrow y=1^4-100=-99 \\
x=-2 &\Rightarrow y=0^4-100=-100 \\
x=-3 &\Rightarrow y=(-1)^4-100=-99 \\
x=-4 &\Rightarrow y=(-2)^4-100=-84 \\
x=-5 &\Rightarrow y=(-3)^4-100=-19 \\
\end{align*}Starting at $x=-6$, the $y$-coordinates obtained in this way are positive. To be sure that there are no more solutions, we can solve the equation $$(x+2)^4-100 < 0,$$which yields $-2-\sqrt[4]{100}<x<-2+\sqrt[4]{100}$ (in decimals, this is roughly $-5.16<x<1.16$). Thus, the graph of $y=(x+2)^4-100$ passes through $\boxed{5}$ points with negative integer coordinates.