根据题意，$\omega$ 是方程 $x^5-1=0$ 的单位根，因此有\[\prod_{k=1}^{4}\left(x-\omega^k\right)=x^4+x^3+x^2+x+1.\]设\[x^2+x+2=(\alpha-x)(\overline\alpha-x),\]其中 $\alpha+\overline{\alpha}=-1,\alpha\overline{\alpha}=2$，则\[\begin{split}P(\omega)P(\omega^2)P(\omega^3)P(\omega^4)&=\prod_{k=1}^4\left(\alpha-\omega^k\right)\cdot \prod_{k=1}^4\left(\overline\alpha-\omega^k\right)\\ &=|\alpha^4+\alpha^3+\alpha^2+\alpha +1|^2\\ &=|-2\alpha^2-1|^2=|2\alpha+3|^2\\&=(2\alpha+3)(2\overline{\alpha}+3)\\
&=11.\end{split}\]