$\Leftrightarrow \sqrt{x^2+15}=6\sqrt x-\sqrt{x^2+3}$$\Rightarrow x^2+15=36x+x^2+3-12\sqrt{x(x^2+3})$
$\Leftrightarrow 1-3x=\sqrt{x(x^2+3)}$
$\Rightarrow 9x^2-6x+1=x^3+3x$
$\Leftrightarrow (x-1)(x^2-8x+1)=0$
$\Leftrightarrow \left[ {\begin{matrix} x=1\\ x=4+\sqrt{15}\\x=4-\sqrt{15}\text{loại} \end{matrix}} \right.$