Step 1:
$\omega=\large\frac{-1+i\sqrt 3}{2}$ and $\omega^3=1$
$\Rightarrow \omega^2=\large\frac{1}{\omega}=\large\frac{-1-i\sqrt 3}{2(\Large\frac{1}{4}+\frac{3}{4})}$
$\Rightarrow \large\frac{-1-i\sqrt 3}{2}$
Step 2:
LHS=$\omega^5+(\omega^2)^5=\omega^3.\omega^2+(\omega^3)^3\omega$
$\Rightarrow \omega^2+\omega=-1$
Hence proved.