Given the contour $C$:
we are asked to calculate $\displaystyle\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z^2-1}dz$. I wrote it as such:
$$\frac{1}{2}\left(\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z-1}dz+\frac{1}{2\pi i}\oint \frac{ze^{z^2-4z}}{z+1}dz\right)$$ and got that it equals $$\frac{1}{2}N(C,1)e^{-3}-\frac{1}{2}N(C,-1)e^5=e^{-3}+e^{5}$$
Does this look correct? I'm a little shaky on whether I got the indexes right, and some of my friends got that it should equal $0.5(e^{-3}+e^{5})$ instead.
According to the picture, it seems that the winding numbers are indeed $+2$ (at $1$), and $-2$ (at $-1$), and the rest of your calculations seem to be correct as well.