<s>[INST] Find all complex roots of $(z+1+i)^4 - 1 + i =0$. Find all complex roots of $(z+1+i)^4 - 1 + i =0$.
Attempt:
I got
\begin{align*}
(z+1+i)^4 &= 1 - i \\
(z + \sqrt{2} e^{i(\frac{\pi}{4})})^4 &= \sqrt{2} e^{i(-\frac{\pi}{4})} \\
z + \sqrt{2} e^{i(\frac{\pi}{4})} &= \sqrt[8]{2} \exp\left(i\frac{-\frac{\pi}{4} + 2k\pi}{4}  \right) \\
z &= \sqrt[8]{2} \exp\left(i\frac{-\frac{\pi}{4} + 2k\pi}{4}  \right) - \sqrt{2}e^{i(\frac{\pi}{4})}
\end{align*}
for $k=0,1,2,3$.
Am I true? If not yet, how to find it?
 [/INST] I got
\begin{align*}
(z+1+i)^4 &= 1 - i \\
(z + \sqrt{2} e^{i(\frac{\pi}{4})})^4 &= \sqrt{2} e^{i(-\frac{\pi}{4})} \\
z + \sqrt{2} e^{i(\frac{\pi}{4})} &= \sqrt[8]{2} \exp\left(i\frac{-\frac{\pi}{4} + 2k\pi}{4}  \right) \\
z &= \sqrt[8]{2} \exp\left(i\frac{-\frac{\pi}{4} + 2k\pi}{4}  \right) - \sqrt{2}e^{i(\frac{\pi}{4})}
\end{align*}
for $k=0,1,2,3$.
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