HMMT 2010 T9

Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$ be a polynomial with complex coefficients such that $a_{i} \neq 0$ for all $i$. Prove that $|r| \leq 2 \max _{i=0}^{n-1}\left|\frac{a_{i-1}}{a_{i}}\right|$ for all roots $r$ of all such polynomials $p$. Here we let $|z|$ denote the absolute value of the complex number $z$.