For which $a$, does $p^2+p+(a-1)$ have only rational solutions?
The roots of $p^2+p+(a-1)$ are $\frac{-1+\sqrt{(5-4a)}}{2}$ and $\frac{-1-\sqrt{(5-4a)}}{2}$, and they are rational iff $\sqrt{5-4a}$ is rational, then iff $5-4a$ is a square of some rational. Then what?
It has only rational solutions if $x^2+x+a-1=(x-r)(x-s)=x^2-(r+s)x+rs$ with rational $r,s$. This says that $a=rs+1=r(-1-r)+1=1-r-r^2$. Hence the quadratic equation has only rational solutions if $a$ is representable by $1-r-r^2$ for some rational $r$. For example, $a=\sqrt{2}$ is not of this form.