$(\sqrt3-2)(\sqrt6+\sqrt2).\sqrt{ \sqrt3+2}$
$=\sqrt2(\sqrt3-2)(\sqrt3+1).\sqrt{\sqrt3+2}$
$=(\sqrt3-2)(\sqrt3+1)\sqrt{4+2\sqrt3}$
$=(\sqrt3-2)(\sqrt3+1).\sqrt{(\sqrt3+1)^2}$
$=(\sqrt3-2)(\sqrt3+1)(\sqrt3+1)$
$=(\sqrt3-1)(\sqrt3+1)(\sqrt3+1)-(\sqrt3+1)(\sqrt3+1)$
$=(3-1)(\sqrt3+1)-(\sqrt3+1)(\sqrt3+1)$
$=(\sqrt3+1)(2-\sqrt3-1)$
$=-(\sqrt3+1)(\sqrt3-1)$
$=-(3-1)$
$=-2$