Factoring a complex number I need to factor this and something isn't making sense to me
$$f(z)=\frac{1}{z^2+2z+2}$$
Using the quadratic, I obtain $z=-1+i$ and $z=-1-i$ which leads me to believe that 
$$f(z)=\frac{1}{z^2+2z+2}\ =\ \frac{1}{(z-1+i)(z-1-i)} $$
But,  when I multiply that back out to check my work, I keep getting $z^2-2z+2$
What am I missing?

$$
z^2+2z+2=(z+1)^2+1=(z+1)^2-i^2=(z+1+i)(z+1-i).
$$
