$\begin{array}{l} \dfrac{{2 - \sqrt 3 }}{{1 + \sqrt {4 + 2\sqrt 3 } }} + \dfrac{{2 + \sqrt 3 }}{{1 - \sqrt {4 - 2\sqrt 3 } }}\\ = \dfrac{{2 - \sqrt 3 }}{{1 + \sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} }} + \dfrac{{2 + \sqrt 3 }}{{1 - \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} }}\\ = \dfrac{{2 - \sqrt 3 }}{{2 + \sqrt 3 }} + \dfrac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }}\\ = \dfrac{{{{\left( {2 - \sqrt 3 } \right)}^2} + {{\left( {2 + \sqrt 3 } \right)}^2}}}{{4 - 3}}\\ = \dfrac{{4 + 3 + 4 + 3 - 4\sqrt 3 + 4\sqrt 3 }}{1} = 14 \end{array}$
