<s>[INST] Maximum and Minimum of complex number $z$ where $|z^4+z^3-z-1| = |z^4-z^3+z-1|$. 
If $z\in C$ satisfies $|z^4+z^3-z-1| = |z^4-z^3+z-1|.$ Then minimum value of $|z-1-2i|$ and maximum value of $|z^2-1-2i|$ is . . .

$\bf{Attempt}$ from $|z^4+z^3-z-1| = |z^4-z^3+z-1|$
$$|(z+1)(z^3-1)| = |(z-1)(z^3+1)|$$
$$\bigg|\frac{z-1}{z+1}\bigg|=\bigg|\frac{z^3-1}{z^3+1}\bigg|=k$$
Could some help me how to solve it, thanks
 [/INST] Hint:  Your factorizations are incomplete.
$$  z^3 - 1 = (z-1)(z^2+z+1)  $$
$$  z^3 + 1 = (z+1)(z^2-z+1)  $$
This leaves $|z^2 -z +1| = |z^2 + z + 1|$ (having already extracted the factors $z-1$ and $z+1$).
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