Given the complex polynomial $P(z) = z^2 + a_1z + a_0$ and the constraint that $|z| > 1$, I'm trying to show that $|P(z)| \geq |z|^2 - |a_1||z| - |a_0|$. The obvious thing to do here of course is to apply the triangle inequality which yields $$ |P(z)| = |z^2 + a_1z + a_0| \leq |z|^2 + |a_1||z| + |a_0| $$ Just as obvious though is the fact that this is not remotely close to what I am trying to show. I'm sure I'm missing a relevant detail. I have attempted, and failed, to productively apply the constraint on $|z|$. This is effectively the start of a proof of the Fundamental Theorem of Algebra.
Would appreciate any tips.