$\sqrt{5+2\sqrt6}+\sqrt{14-4\sqrt6}$
$=\sqrt{3+2\sqrt2.\sqrt3+2}+\sqrt{12-2.2\sqrt3.\sqrt2+2}$
$=\sqrt{(\sqrt3)^2+2\sqrt3.\sqrt2+(\sqrt2)^2}+\sqrt{(2\sqrt3)^2-2.(2\sqrt3).\sqrt2+(\sqrt2)^2}$
$=\sqrt{(\sqrt3+\sqrt2)^2}+\sqrt{(2\sqrt3-\sqrt2)^2}$
$=|\sqrt3+\sqrt2|+|2\sqrt3-\sqrt2|$
$=\sqrt3+\sqrt2+2\sqrt3-\sqrt2$
$=3\sqrt3$