jmc

We can write the equation as $$(x+\sqrt{2}{)}^3+(x+\sqrt{2})=0.$$Then $$(x+\sqrt{2})[(x+\sqrt{2}{)}^2+1]=0,$$so $x=-\sqrt{2}$ or $(x+\sqrt{2}{)}^2=-1.$ For the latter equation, $$x+\sqrt{2}=\pm i,$$so $x=-\sqrt{2}\pm i.$

Thus, the solutions are $\boxed{-\sqrt{2},-\sqrt{2}+i,-\sqrt{2}-i}.$