Does there exist a quadratic polynomial $f \in K(x)[T]$ and $\Delta\in K[x]$ such that $f^2 + \Delta$ and $x^2-1+\Delta$ are both squares

Question

Does there exist a number field $K$, not containing $i$, a quadratic polynomial $f\in K(x)[T]$, and an arbitrary polynomial $\Delta\in K[x]$ such that $f^2 + \Delta$ and $x^2-1 + \Delta$ are both squares?