\( (\sqrt{2}+\sqrt{2+\sqrt{3}})^{2}+(1-\sqrt{4+2 \sqrt{3}})^{2} \)
\(= 2 + 2*\sqrt{2}*\sqrt{2+\sqrt{3}} + 2+\sqrt{3} + 1 - 2* \sqrt{4+2 \sqrt{3}}+ 4+2 \sqrt{3} \)
\(= 9 + 2*\sqrt{2}*\sqrt{2+\sqrt{3}} - 2* \sqrt{4+2 \sqrt{3}}+3 \sqrt{3} \)
\(= 9 + 2\sqrt{4+2\sqrt{3}} - 2* \sqrt{4+2 \sqrt{3}}+3 \sqrt{3} \)
\(= 9 +3 \sqrt{3} \)