Number of roots in the first quadrant I want to find how many roots of the equation $z^4+z^3+1=0$ lies in the first quadrant.
Using Rouche's Theorem how to find ?

For completeness, here I present a more standard solution to this problem:
Let $p(z):=z^4+z^3+1$. Consider the following path

Since $p(x)>0$ and $p(ix)=x^4+1-ix^3\neq 0$ for $x\geq 0$, we deduce that $p$ has no roots on $\gamma_R$. Therefore, we can apply the Argument Principle to obtain that the number of roots of $p$ in the first cuadrant is the winding number of $0$ around $p\circ \gamma_R$, for sufficiently big $R$.
Besides, for big $R$ the argument variation of $p\circ \text{quarter circle}$ is approximately $\Delta \arg z^4=4\frac{\pi}{2}=2\pi$. Therefore, this fact combined with $p(x)>0$, $p(ix)\in 4\text{th quadrant}$ tells us that $p$ has exactly one root in the first quadrant.
