The common roots of the equations

${\mathrm{Z}}^3+2{\mathrm{Z}}^2+2\mathrm{Z}+1=0$ and

${\mathrm{Z}}^{2018}+{\mathrm{Z}}^{2017}+1=0$ satisfy the equation

$$\begin{gathered} {\mathrm{z}}^3+2{\mathrm{z}}^2+2\mathrm{z}+1=0 \\ \Rightarrow ({\mathrm{z}}^3+1)+2\mathrm{z}(\mathrm{z}+1)=0 \\ \Rightarrow (\mathrm{z}+1) ({\mathrm{z}}^2-\mathrm{z}+1)+2\mathrm{z}(\mathrm{z}+1)=0 \\ \Rightarrow (\mathrm{z}+1) ({\mathrm{z}}^2+\mathrm{z}+1)=0 \\ \therefore \mathrm{z}=-1, \mathrm{z}=\mathrm{w}, \mathrm{z}={\mathrm{w}}^2 \\ and {\mathrm{z}}^{2018}+{\mathrm{z}}^{2017}+1=0 \mathrm{satisfied} \mathrm{by} \mathrm{z}=\mathrm{w}, {\mathrm{w}}^2 \\ \boxed{\therefore \mathrm{w}, {\mathrm{w}}^2 \mathrm{also} \mathrm{satisfy} {\mathrm{z}}^4+{\mathrm{z}}^2+1=0} \end{gathered}$$