We have $a = 1, h = - \sqrt{3} , b = 3$,
$ g = -\frac{3}{2} , f = \frac{3 \sqrt{3}}{2} , c = - 4$ .
Thus $abc + 2fgh - af^2 - bg^2 - ch^2 = 0$
Hence the equation represents a pair of straight lines.
Again $\frac{a}{h} = \frac{h}{b} = \frac{g}{f} = - \frac{1}{\sqrt{3}}$
$\therefore $ the lines are parallel. The distance between them
$= 2 \sqrt{\frac{g^{2} -ac}{a\left(a+b\right)}} = 2 \sqrt{\frac{\frac{9}{4} +4}{1\left(1+3\right)}} = \frac{5}{2} $