<s>[INST] Show that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$. I need help with showing that $z^3 + (1+i)z - 3 + i = 0$ does not have any roots in the unit circle $|z|\leq 1$?
My approach so far has been to try to develop the expression further.
$$ z^3 +(1+i)z-3+i = z(z^2+i+1)-3+i$$
$$z(z^2+i+1) = 3 - i \longrightarrow |z(z^2+i+1)| = |3 - i|$$
This gives me the expression:
$z((z^2+1)^2+(1)^2) = \sqrt{10}$
Which can be written as:
$z(z^4 +2z^2 +2) = \sqrt{10}$
But how do I move on from here? Or am I attempting the wrong solution?
Thank you for your help!
 [/INST] $z^3 + (1+i)z - 3 + i = 0\iff z^3+(1+i)z=3-i$
Now, If $|z|\leq 1$, then $|z^3+(1+i)z|\leq |z|^3+|1+i||z|\leq1+\sqrt2$
As $|3-i|=\sqrt{10}\gt 1+\sqrt{2}$
Therefore, $z^3+(1+i)z\neq 3-i$ for any $z\in \Bbb C, |z|\leq 1$
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