Write the function $\frac{1}{(z+1)(3-z)}$ as a Laurent series. $$f(z)=\frac{1}{(z+1)(3-z)}=\frac{1}{4z+4} + \frac{1}{12-4z}$$
$$\frac{1}{4z+4}=\frac{1}{4z}\frac{1}{1-\frac{-1}{z}}=\frac{1}{4z}\sum_{k=0}^{\infty} \left(\frac{-1}{z}\right)^k$$
$$\frac{1}{12-4z}=\frac{1}{12}\frac{1}{1-\frac{z}{3}}=\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{z}{3}\right)^k$$
$$f(z)=\frac{1}{4z}\sum_{k=0}^{\infty} \left(\frac{-1}{z}\right)^k+\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{z}{3}\right)^k$$
I can rewrite that as $$f(z)=\frac{1}{4z}\sum_{k=-\infty}^{0} (-1)^k z^k+\frac{1}{12}\sum_{k=0}^{\infty} \left(\frac{1}{3}\right)^k z^k$$.
I need to move the $\frac{1}{4z}$ and $\frac{1}{12}$ into the sums but finding a series that will converge to each, but I have no idea what to use for either. Any suggestions? Am I taking a wrong approach or is there an obvious series to use for this?
Edit: The center is $0$ and the region in $1 \le |z| \le 3$.
I think I can use a geometric sequence to say $\frac{1}{4z}=\sum_{k=0}^{\infty}\frac{1}{2}(\frac{1}{2})^{k-1}\frac{z}{k}$ and $\frac{1}{12}=\sum_{k=0}^{\infty}\frac{1}{36}(\frac{2}{9})^{k-1}\frac{z}{k}$. I'm pretty sure that's true, but it seems like it makes the whole thing a complicated mess. 

I can write the expression as:
$$\frac{1}{(z+1)(-z+3)} = \frac{-1}{(z+1)(z-3)}$$
Use partial fractions:
$$\frac{-1}{(z+1)(z-3)} = \frac{A}{z+1}+\frac{B}{z-3}$$
$$-1 = A(z-3)+B(z+1)$$
If z=3, then
$$-1 = B(3+1)$$
$$-1 = B(4)$$
$$B=\frac{-1}{4}$$
If z=1, then 
$$-1 = A(1-3)$$
$$-1 = A(-2)$$
$$1 = A(2)$$
$$A = 1/2$$
So, we have:
$$\frac{-1}{(z+1)(z-3)} = (\frac{1}{2})\frac{1}{z+1}+(\frac{-1}{4})\frac{1}{z-3}$$
The previous expression is a Laurent Series. Because the expression has no expressions of the form
$$\frac{1}{z^{n}}$$
The Laurent Series behaves as a Taylor Series.
If we suppose that in z=0 the expression is not analytic, then we can write it as
$$\frac{-1}{(z+1)(z-3)} = (\frac{1}{2})\frac{1}{z(1+\frac{1}{z})}+(\frac{-1}{4})\frac{1}{z(1-\frac{3}{z})}$$
