Show that any root $z$ of $z^4 + z + 3 = 0$ satisfies $|z| > 1$ Show that any root $z$ of $z^4 + z + 3 = 0$ satisfies $|z| > 1$ and that any root $z$ of $4z^4 + z + 1 = 0 $ satisfies $ |z| \le 1.$
What I've done so far:
$|z^4+z+3| \le |z|^4+|z|+3$.
If $0<|z|\le1, $ then $3< |z|^4+|z|+3 \le 5$. 
How do I then complete the proof?

If $|z|\leq 1$ then $|z^4+z|= |z||z^3+1|\leq|z^3+1|\leq |z|^3+1\leq 1+1=2$.
So, if $|z|\leq 1$ then $|z^4+z|\leq 2$ and hence $z^4+z+3\neq 0$.
