\(2+\sqrt2=a(1+\sqrt2)+3\)
\(\Leftrightarrow a(1+\sqrt2)=2+\sqrt 2-3\)
\(\Leftrightarrow a(1+\sqrt 2)=\sqrt 2-1\)
\(\Leftrightarrow a=\dfrac{\sqrt 2-1}{1+\sqrt 2}=\dfrac{(\sqrt 2-1)(1-\sqrt 2)}{(1+\sqrt2)(1-\sqrt2)}\)
\(=\dfrac{(\sqrt 2-1)(1-\sqrt 2)}{1-2}=-(\sqrt 2-1)(1-\sqrt 2)\)
\(=(\sqrt 2-1)(\sqrt 2-1)=(\sqrt 2-1)^2\)
\(=3-2\sqrt2\).
