The book is right. Check you have got the right singularity.
With "right" i mean the one who is inside your curve and apply the theorem as you just did.
Oh, because the curve is |z+1|= 2, a = -2 is inside the curve, and a = 2 is not. Therefore, I get $\displaystyle \int \! \frac{z^2 (\frac{1}{2 - z}) \, dz}{z - (-2)}$