Factoring a complex number

Question

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?

Answer 1

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