Triangle inequality in complex numbers let $|z|=R$, by using the triangle inequality, find a lower bound for $$|z^4+5z^2+4|$$
approachh:  $$|z^4+5z^2+4|\geq|z^4|-|5z^2+4| \geq |z^4|-(|5z^2|+4)=|z^4|-|5z^2|-4=R^4-5R^2-4$$
but the solution is $$|z^4+5z^2+4| \geq R^4-5R^2+4$$
what went wrong?

You get a sharper lower bound if you render
$z^4+5z^2+4=(z^2+1)(z^2+4)$
and find the lower bounds for the absolute values of the factors on the right side.
