What is the inverse Z transform of this:
You just need to multiply the terms and sort by powers of $z$. If you do so you should get

$$X(z)=\frac{z^2}{4}-\frac54+z^{-2}\tag{1}$$

So the work is in multiplying out the terms, not in actually figuring out the inverse $\mathcal{Z}$-transform, because from $(1)$ you can immediately write down $x[n]$:

$$x[n]=\frac14\delta[n+2]-\frac54\delta[n]+\delta[n-2]\tag{2}$$