How to calculate $\frac{1}{2i\pi}\int_c|1+z+z^2|^2dz$?

Question

Let c denotes the unit circle centered at the origin in C then $\frac{1}{2i\pi}\int_c|1+z+z^2|^2dz$ where the integral is taken anti clockwise along C equals

1. 0

2. 1

3. 2

4. 3

   I tried this by considering the properties of the complex numbers $|z|^2=z\bar{z}$, by considering $z=Re^{i\theta}$ and by substituting $z=x+iy$ but i didn't get how to solve this

Answer 1

Note that $z\bar z=|z|^2$. Expanding $$ (1+z+z^2)(1+\bar z+\bar z^2) $$ one gets $$ (1+\bar z+\bar z^2) +(z+|z|^2+z|z|^2) +z^2+ z|z|^2+|z|^4. $$ Now let $z=e^{i\theta}$ ($z$ is on the unit circle!) and use the linearity of integrals.