How many complex numbers have $|z|<1$ such that $az ^ 4 + z ^ 3 + bz ^ 2 + cz + d = 0$

Question

Let the four positive numbers $a, b, c, d$ satisfy $a + b + c + d <1$. Ask how many complex numbers have $|z|<1$ such that $$az ^ 4 + z ^ 3 + bz ^ 2 + cz + d = 0$$

Answer 1

Hint

If $g(z)=z^3$, we can appeal to Rouché's theorem since for $|z|=1$ $$|f(z)-g(z)| = |az^4+bz^2+cz+d|\le a+b+c+d <1 <|f(z)| + |g(z)|$$