Determining Laurent Series

This question is from Complex Variables and Applications by Brown & Churchill, 8ed. Section 62, #2.

Determine the Laurent Expansion of:


for the interval, $ 0 < |z + 1| < \infty$. How do I go about doing that?

Here’s what I have so far. If I substitute $ w = (z + 1) \implies w^2 = (z+1)^2$ then I get

$$\begin{aligned} \exp(z) \sum_{k = 0}^{\infty} \frac{1}{w^2} &= \exp(z) \sum_{k=0}^{\infty}(-1+w)^n(-1)^n(1+n)\\ &= \exp(z)\sum_{k =0}^{\infty}(z^n)(-1)^n(1+n)\end{aligned}$$

But I don’t know if this is the correct approach or if this will even yield a viable Laurent Series. Any ideas?

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