I'm having trouble progressing through (in my experience) the tedious calculations required to obtain a Laurent Expansion of a complex function.

The problem arises in finding the series within the annulus. Thank you in advanced for your help.
I'm having trouble progressing through (in my experience) the tedious calculations required to obtain a Laurent Expansion of a complex function.

The problem arises in finding the series within the annulus. Thank you in advanced for your help.
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Hint for (a): where are the singularities of $f(z)$?
Hint for (b): partial fractions.