\[\begin{array}{l}
\sqrt {x + 2\sqrt {2x - 4} } + \sqrt {x - 2\sqrt {2x - 4} } \left( {x \ge 2} \right)\\
= \frac{1}{{\sqrt 2 }}\left( {\sqrt {2x + 4\sqrt {2x - 4} } + \sqrt {2x - 4\sqrt {2x - 4} } } \right)\\
= \frac{1}{{\sqrt 2 }}\left( {\sqrt {2x - 4 + 4\sqrt {2x - 4} + 4} + \sqrt {2x - 4 - 4\sqrt {2x - 4} + 4} } \right)\\
= \frac{1}{{\sqrt 2 }}\left( {\sqrt {{{\left( {\sqrt {2x - 4} + 2} \right)}^2}} + \sqrt {{{\left( {\sqrt {2x - 4} - 2} \right)}^2}} } \right)\\
= \frac{1}{{\sqrt 2 }}\left( {|\sqrt {2x - 4} + 2| + |\sqrt {2x - 4} - 2|} \right)\\
= \frac{1}{{\sqrt 2 }}\left( {\sqrt {2x - 4} + 2 + \sqrt {2x - 4} - 2} \right)\left( {do:x \ge 2 = > \sqrt {2x - 4} + 2;\sqrt {2x - 4} - 2 \ge 0} \right)\\
= \frac{1}{{\sqrt 2 }}\left( {2\sqrt {2x - 4} } \right)\\
= \sqrt 2 .\sqrt {2x - 4} \\
= \sqrt {4x - 8}
\end{array}\]