Let $\mathrm{\alpha },\mathrm{\beta }$ be the roots of ${\mathrm{x}}^2+\mathrm{x}+1=0$ then the equation whose roots are ${\mathrm{\alpha }}^{229}$ and ${\mathrm{\alpha }}^{1004}$ is

$\begin{array}{l}If\alpha and\beta areof{x}^2+x+1=0then\\ x=\frac{-1\pm \sqrt{1-4}}{2}=\frac{-1\pm i\sqrt{3}}{2}\\ \therefore \alpha =\omega ,\beta ={\omega }^2\\ \therefore {\alpha }^{229}+{\alpha }^{1004}={\omega }^{229}+{\omega }^{1004}=\omega +{\omega }^2=-1\\ {\alpha }^{225}.{\alpha }^{1004}=\omega .{\omega }^2={\omega }^3-1\\ \therefore \mathrm{Re}quiredEquationis{x}^2-({\alpha }^{229}+{\alpha }^{1004})x+{\alpha }^{229}.{\alpha }^{1004}=0\\ \Rightarrow x^2+x+1=0\end{array}$