I'm trying to show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $$ |z|=\max\{1, |a|+|b|+|c|\}. $$ The progress I've reached so far is merely $$ |z|=r;\quad 0\le |a|+|b|r+|c|r^2+r^3. $$ I don't understand where the "$1$" comes from and how I can derive $|a|+|b|+|c|$ from the equation.
Update: I got this $$ r^3\le |a|+|b|r+|c|r^2. $$
and it is clear where $1$ comes from. But the case when $r>1$ is still not very clear to me.