#### Answer

The answer is $x=\sqrt5(2\sqrt3+3)\div30$.

#### Work Step by Step

Given:
$4\sqrt15/(1+\sqrt3)=(1+\sqrt3)/x$
By Cross multiplication we get,
$4\sqrt15*x=(1+\sqrt3)*(1+\sqrt3)$
$4x\sqrt15=(1+\sqrt3)^{2}$
$4x\sqrt3*\sqrt5=(1^{2}+\sqrt3^{2}+2*1*\sqrt3)$
$4x\sqrt3*\sqrt5=(1+3+2\sqrt3)$
$x=(4+2\sqrt3)/4*\sqrt3*\sqrt5$
$x=(2+\sqrt3)/2\sqrt15$
(Dividing by 2 in the numerator and denominator)
$x=(2+\sqrt3)/2\sqrt15*\sqrt15/\sqrt15$
(Multiply and divide by $\sqrt15$)
$x=(2+\sqrt3)/2\sqrt15*\sqrt15/\sqrt15$
$x=(2+\sqrt3)*\sqrt15/(2*15)$
$x=(2\sqrt15+\sqrt3*\sqrt15)/30$
$x=(2*\sqrt3*\sqrt5+3*\sqrt5)/30$
$x=\sqrt5(2\sqrt3+3)\div30$