Complex integration around unit circle centresd at origin

Question

I am trying masters entrance exam question papers and I am unable to solve this particular question, so I am asking it here.

Let C denote the unit circle centred at origin in $\mathbb{C} $ then find value of

$$\frac{1}{2\pi i}\int_{C} |1+z+z^2|^2 dz$$.

Attempt: Integration of any analytic function along a closed curve is 0 but the function is not analytic , also as ${|z|}^2 $ is analytic anywhere except 0 so I can't use residue theorem.

So, I am struck and kindly give hint!!

Answer 1

Hint: The integral can be written as $$\frac 1 {2\pi i} \int_0^{2\pi} (1+e^{i\theta}+e^{2i\theta})( \overline {1+e^{i\theta}+e^{2i\theta}}) ie^{i\theta} d\theta.$$ Just expand the product and integrate term by term.