What is the Laurent series for a function $f$ defined in an annulus $A$ ? Find the Laurent series for $f(z)=\frac{10}{(z+2)\left(z^{2}+1\right)}$ on the annuli

$$\begin{aligned} &A_{1}=\{z \in \mathbb{C}|0<| z \mid<1\} \quad \text { and } \\ &A_{2}=\{z \in \mathbb{C}|1<| z \mid<2\} \end{aligned}$$