Roots of a complex equation are outside the unit disc using the triangle inequality Problem: Use the triangle inequality to show that the roots of the complex equation $$z^4+z+4=0$$ has roots all outside the unit disc $|z|\le1$
My Thought Process: Clearly I need to use the triangle inequality and this would be a proof by contradiction. Assuming that $|z|\le1$, then the triangle inequality gives $|z^4+z|\le|z^4|+|z|\le2$ but I'm not sure where my contradiction would be.

Note that $z^4+z+4=0$ implies that $-4=z+z^4$, and, by the triangle inequality,
$$4=|-4|=|z+z^4|\leq |z|+|z|^4$$
Hence if $|z|=r$ then $4\leq r+ r^4$. Is it possible that $r\leq 1$?
By the same inequality, it turns out that $r$ has to be greater than $1.28$ because $1.28+(1.28)^4\approx 3.96<4$.
