Let

a = \sqrt{25+10\sqrt{6}}

b =\sqrt{ 25-10\sqrt{6}}

Then

ab = \sqrt{25^2-10^2\cdot 6}

ab = 5

a^2+b^2 = 50

(a+b)^2 = a^2+b^2+2ab = 60

a+b = \sqrt{60}=2\sqrt{15}

Therefore,

\sqrt{25+10\sqrt{6} } +\sqrt{25-10\sqrt{6} } =2\sqrt{15}