[tex]\displaystyle\\
\frac{1}{2+ \sqrt{5} } + \frac{1}{2- \sqrt{5} }=\\\\
= \frac{1}{\sqrt{5}+2 } + \frac{1}{ -(\sqrt{5}-2) }=\\\\
=\frac{1}{\sqrt{5}+2 } - \frac{1}{\sqrt{5}-2 }=\\\\
~~~~~~~~~~~~~~~~~~~~~\text{Numitorul comun este: }~ (\sqrt{5}+2 )(\sqrt{5}-2 )\\\\
= \frac{1\times (\sqrt{5}-2)}{(\sqrt{5}+2 )(\sqrt{5}-2)} - \frac{1\times (\sqrt{5}+2)}{(\sqrt{5}-2 )(\sqrt{5}+2)}=\\\\
= \frac{\sqrt{5}-2}{(\sqrt{5})^2-2^2 } - \frac{\sqrt{5}+2}{(\sqrt{5})^2-2^2}=
[/tex]
[tex]\displaystyle\\
= \frac{\sqrt{5}-2}{5-4 } - \frac{\sqrt{5}+2}{5-4}=\\\\
= \frac{\sqrt{5}-2}{1} - \frac{\sqrt{5}+2}{1}=\\\\
=\sqrt{5}-2 -(\sqrt{5}+2)=\\\\
=\underline{\sqrt{5}}-2 -\underline{\sqrt{5}}-2= -2 -2 = \boxed{\bf -4}
[/tex]
