[tex]\sqrt{2+3\sqrt{2}} = \sqrt{a}+\sqrt{b} \Big|^2[/tex]
[tex]\Rightarrow 2+3\sqrt{2} = a+2\sqrt{ab}+b[/tex]
[tex]\Rightarrow 2+\sqrt{3^2\cdot 2} = a+b+\sqrt{2^2\cdot ab}[/tex]
[tex]\Rightarrow 2+\sqrt{18} = a+b+\sqrt{4ab}[/tex]
[tex]\Rightarrow \begin{cases} a+b = 2\\ 4ab=18\end{cases} \Rightarrow \begin{cases} a+b = 2\\ ab=\dfrac{9}{2}\end{cases} \Rightarrow t^2-2t+\dfrac{9}{2} = 0[/tex]
[tex]\Rightarrow 2t^2-4t+9 = 0[/tex]
[tex]\Delta = (-4)^2-4\cdot 2\cdot 9 = 16-72 = -56 < 0[/tex]
[tex]\Rightarrow a,b \notin \mathbb{R}[/tex]
