Expand the function
$$ f(z) = \frac {2(z + 2)} {z^2 − 4z + 3} $$
in a Taylor series about the point $ z = 2 $ and find the circle C inside of which the series converges. Find a Laurent series that converges in the region outside of C.

I tried writing the denominator as $ (z-3)(z-1) $ to find the singularities $ z = 1,3 $ both simple. There exists a circle around $ z = 2 $ such that $f(z) $ is analytic so we can write it as a Taylor series. I got stuck trying to make the function look like a known Taylor series like $e^z$ or $\frac {1} {1-z}$