Challenge for Polynomial with Complex Coefficients

[anemone]

Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region

$|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.

 

[anemone]

Hint:

Spoiler

Consider $|\sqrt{b^2-4ac}|$.