Let us suggest that there exist rational $a, b$, such that: $a = x + \sqrt{2}$ and $b = x^4 + \sqrt{2}$. Thus $x = a - \sqrt{2}$. After substitution in another equality we obtain:
$$b = a^4 - 4a^3\sqrt{2} + 12a^2 - 8a\sqrt{2} + 4 + \sqrt{2}.$$
Right part of equality has to be rational, thus the sum of components that contain $\sqrt{2}$ has to equal 0. It is enough to answer the question, whether there exist rational solution of $4a^3 + 8a - 1 = 0$. It is not difficult to check that there is no solutions. We have a contradiction.