Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in $\mathbf{Q}$?
This question can be phrased also as follows:
Are there infinitely many rational numbers $(a,b)$ such that the extensions $\mathbf{Q}(\sqrt{a^3+1})$ and $\mathbf{Q}(\sqrt{b^3+2})$ are quadratic and equal?