Explanation

(c)
Given equation $x^{2} + 2x + 2xy + my - 3 = 0$ can be rewritten as $x^{2} + 2x(1 + y) + (my - 3) = 0$.
But factors are rational so discriminant $b^{2} - 4ac$ is a perfect square.
Now, $b^{2} - 4ac = 4{\{(1 + y)}^{2} - (my - 3)\} \geq 0$
$\Rightarrow 4\{ y^{2} + 1 + 2y - my + 3\} \geq 0$
$\Rightarrow y^{2} + 2y - my + 4 \geq 0$
Hence, $2y - my = \pm 4y$(as it is perfect square).
$\Rightarrow 2y - my = 4y$
$\Rightarrow m = - 2$
Now, taking $( - )$ve sign, we get $m = 6$