Answer
The $-4q$ charge should be placed at the position $x = 6.0~mm$
Work Step by Step
To find the distance $r$ to the right of $x = 2.0~mm$ where the $-4q_1$ should be placed, we can equate the magnitude of the electric field due to each charge:
$\frac{1}{4\pi~\epsilon_0}~\frac{q}{(0.0020~m)^2} = \frac{1}{4\pi~\epsilon_0}~\frac{4q}{r^2}$
$\frac{1}{(0.0020~m)^2} = \frac{4}{r^2}$
$r^2 = (4)(0.0020~m)^2$
$r = (2)(0.0020~m)$
$r = 4.0~mm$
The $-4q$ charge should be placed at the position $x = 6.0~mm$
