$\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}}$
$=\sqrt{3+2.\sqrt{2}.\sqrt{3}+2}+\sqrt{3-2.\sqrt{2}.\sqrt{3}+2}$
$=\sqrt{(\sqrt{3}+\sqrt{2})^2}+\sqrt{(\sqrt{3}-\sqrt{2})^2}$
$=|\sqrt{3}+\sqrt{2}|+|\sqrt{3}-\sqrt{2}|$
$=\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}$
$=2\sqrt{3}$