Are there infinitely many pairs of rational numbers such that... 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?

Yes.  Consider the two elliptic curves:
$$7r^2 = a^3+1$$
and
$$7s^2 = b^3+2.$$
One can check (using SAGE or MAGMA) that these two curves have infinitely many rational points (in fact they both have rank 1).
For rational points $(a,r)$ and $(b,s)$ on these curves we have that $a$ and $b$ satisfy the requirements of your question.
