#### Answer

x = $\frac{ 2 \sqrt 15 + \sqrt 45}{30}$

#### Work Step by Step

$\frac{4\sqrt 15}{1 + \sqrt 3}$ = $\frac{1 + \sqrt 3}{x}$
We cross multiply
$4 \sqrt 15 (x) = (1 + \sqrt 3)(1 + \sqrt 3)$
We divide both sides by $4 \sqrt 15$
x = $\frac{1 + 2 \sqrt 3 + 3}{4 \sqrt 15}$
We add 3 and 1 together
x = $\frac{2 \sqrt 3 + 4}{4 \sqrt 15}$
We divide the coefficients by 2
x = $\frac{ \sqrt 3 + 2}{2 \sqrt 15}$
We multiply it by the conjugate which is $\sqrt 15$
x = $\frac{ \sqrt 3 + 2}{2 \sqrt 15} \times \frac{\sqrt 15}{\sqrt 15} $
We multiply the numerator and denominator to simplify
x = $\frac{ \sqrt 3 + 2 \times \sqrt 15}{2 \sqrt 15 \times \sqrt 15}$
x = $\frac{ 2 \sqrt 15 + \sqrt 45}{30}$