$\begin{aligned} & \left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2 \\ & =\left(3+3 \omega+3 \omega^2+2 \omega\right)^2+\left(3+3 \omega+3 \omega^2+2 \omega^2\right)^2 \\ & =\left[3\left(1+\omega+\omega^2\right)+2 \omega\right]^2+\left[3\left(1+\omega+\omega^2\right)+2 \omega^2\right]^2 \\ & =(3(0)+2 \omega)^2+\left[3(0)+2 \omega^2\right]^2=4 \omega^2+4 \omega^4 \\ & =4 \omega^2\left(1+\omega^2\right)=4 \omega^2(-\omega) \\ & =-4 \omega^2=-4\end{aligned}$