Given: Eqn. of the plane $4x+y+z+2=0$ ......(i)
Eqn. of the line is $\large\frac{x}{1}=\frac{y}{-2}=\frac{z-5}{1}$$=\lambda$.....(ii)
A line parallel to the plane (i) is $\perp$ to the normal, $ \overrightarrow n=(4,1,1)$ to (i)
A line $\perp$ to (ii) and $\overrightarrow n$ has $d.r. = \overrightarrow n\times (1,-2,1)$
(Since a vector $\perp$ to $\overrightarrow a\:\:and\:\:\overrightarrow b$ is $\overrightarrow a\times\overrightarrow b$)
$(4,1,1)\times (1,-2,1)=(3,-3,9)$
$\therefore$ $d.r.$ of the required line is $(-3,3,9)=(-1,1,3)$
$\therefore$ The eqn, of the required line through $(2,-1,-1)$ is
$\large\frac{x-2}{-1}=\frac{y+1}{1}=\frac{z+1}{3}$