Plotting the set of complex numbers that satisfy an inequality

Question

I am trying to understand how to plot the set on the complex plane that satisfies $$\mid 1 + z + z^2|<4$$ where $z \in \mathbb{C}$. I have tried to use $z = a+bi$ and substitute in to the inequality, but I am left with imaginary numbers that I do not know how to interpret. Is there anyway to visualize this inequality graphically on the complex plane? Thanks in advance.

Answer 1

Note that the roots of $z^2+z+1$ are $-\frac12\pm\frac{\sqrt3}2i$. So$$\lvert z^2+z+1\rvert<4\iff\left\lvert z-\left(-\frac12+\frac{\sqrt3}2i\right)\right\rvert\times\left\lvert z-\left(-\frac12-\frac{\sqrt3}2i\right)\right\rvert<4.$$Therefore, you have the set of those points $z\in\mathbb C$ such that the product of their distances to $-\frac12+\frac{\sqrt3}2i$ and to $-\frac12-\frac{\sqrt3}2i$ is smaller that $4$. So, you get the region of $\mathbb C$ bounded by a Cassini oval: