<s>[INST] Factored $z^4 + 4$ into $(z^2 - 2i)(z^2 + 2i)$. How to factor complex polynomials such as $z^2 - 2i$ and $z^2 + 2i$? I was wondering how to factor complex polynomials such as  $z^2 - 2i$ and $z^2 + 2i$?
I originally had $z^4 + 4$, which I factored to $(z^2 - 2i)(z^2 + 2i)$ by substituting $X = z^2$ and using the quadratic formula. However, I'm not sure if there exists any method to factor $z^2 - 2i$ and $z^2 + 2i$? 
I would greatly appreciate it if people could please explain how one would go about this.
 [/INST] For example:
$$z^2-2i=z^2-(1+i)^2=(z-1-i)(z+1+i).$$
I think the following a bit of better.
$$z^4+4=z^4+4z^2+4-4z^2=(z^2+2)^2-(2z)^2=$$
$$=(z^2-2z+2)(z^2+2z+2)=((z-1)^2-i^2)((z+1)^2-i^2)=$$
$$=(z-1-i)(z-1+i)(z+1-i)(z+1+i).$$
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