Three electric charges 2q , 2q , -4q are placed at the corners of...

The system of charges can be considered as two dipoles by splitting the $-4q$ charge into two charges of $-2q$ each.

One dipole is formed by $-2q$ and $2q$ along one side of the equilateral triangle, having a dipole moment $p_1 = (2q)x = 2qx$.

The second dipole is formed by the other $-2q$ and $2q$ along the adjacent side, having a dipole moment $p_2 = (2q)x = 2qx$.

The angle between the two dipole moment vectors is $60^{{}^{\circ}}$, which is the interior angle of the equilateral triangle.

The magnitude of the resultant dipole moment is given by:

$p_{net} = \sqrt{p_1^2 + p_2^2 + 2p_1p_2\cos 60^{{}^{\circ}}}$

Substituting $p_1 = p_2 = 2qx$ and $\cos 60^{{}^{\circ}} = \dfrac{1}{2}$:

$p_{net} = \sqrt{(2qx)^2 + (2qx)^2 + 2(2qx)(2qx)\left(\dfrac{1}{2}\right)}$

$p_{net} = \sqrt{4q^2x^2 + 4q^2x^2 + 4q^2x^2}$

$p_{net} = \sqrt{12q^2x^2} = 2\sqrt{3}qx$

Answer: $2\sqrt{3}qx$