12+3+25−3+12−5=1(2−3)(2+3)(2−3)+2(5+3)(5−3)(5+3)+1(2+5)(2−5)(2+5)=2−34−3+25+235−3+2+54−5=2−31+2(5+3)2+2+5−1=2−3+5+3−2−5=0

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$\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}} =$

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Detailed Solution

$\begin{array}{l} \frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}} \\ = \frac{1 (2 - \sqrt{3})}{(2 + \sqrt{3}) (2 - \sqrt{3})} + \frac{2 (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + 3)} + \frac{1 (2 + \sqrt{5})}{(2 - \sqrt{5}) (2 + \sqrt{5})} \\ = \frac{2 - \sqrt{3}}{4 - 3} + \frac{2 \sqrt{5} + 2 \sqrt{3}}{5 - 3} + \frac{2 + \sqrt{5}}{4 - 5} \\ = \frac{2 - \sqrt{3}}{1} + \frac{2 (\sqrt{5} + \sqrt{3})}{2} + \frac{2 + \sqrt{5}}{- 1} \\ = 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - 2 - \sqrt{5} = 0 \end{array}$