Shorten right side : \(\sqrt[3]{45+29\sqrt{2}}-\sqrt[3]{40+14\sqrt{2}}\)

\(=\sqrt[3]{2\sqrt{2}+3\left(\sqrt{2}\right)^2\cdot3+3\sqrt{2}\cdot9+27}-\sqrt[3]{2\sqrt{2}+3\left(\sqrt{2}\right)^2\cdot2+3\sqrt{2}\cdot4+8}\)

\(=\sqrt[3]{\left(\sqrt{2}+3\right)^3}-\sqrt[3]{\left(\sqrt{2}+2\right)^3}=\sqrt{2}+3-\left(\sqrt{2}+2\right)=1\)

So the expression is equal : \(\sqrt{2x^2+1}+\sqrt{x^2-3x+\dfrac{17}{2}}=1\)

\(\Leftrightarrow\sqrt{2x^2+1}+\sqrt{\left(x-\dfrac{3}{2}\right)^2+\dfrac{25}{4}}=1\)

The expression has no solution cause the left side \(>\sqrt{2x^2+1}\ge1\).

Thank you !