Sum of common roots of the equations $z^3+2z^2+2z+1=0$ and $z^{1985}+z^{100}+1=0$ is

$z^3+2z^2+2z+1=z^3+1+2z(z+1)$

$=(z+1)(z^2+z+1)$

$\Rightarrow z=-1,\omega ,{\omega }^2$

Since $z=-1$, does not satisfy $z^{1985}+z^{100}+1=0$

$z=\omega ,{\omega }^2$satisfy the equation $z^{1985}+z^{100}+1=0$

$\therefore$ sum of common roots $=\omega +{\omega }^2=-1$