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.

If $z^4+z+4=0$, then $z^4+z=-4$, so you'd need $\lvert z^4+z \rvert = 4 $. But you've shown that $\lvert z^4+z \rvert \leq 2$ for $\lvert z \rvert \leq 1$.
