Hint 4

Consider a small element of the ring of length $a\,d\theta$ at an angle $\theta$ from the line between the centre of the ring and the point mass. The distance from this element and the point mass is $r=\sqrt{x^2+a^2 -2ax\cos\theta}$. Hence, if $\lambda=M/(2\pi a)$ is the linear density of the ring, the element contributes

$$dU=-{Gm\lambda a\,d\theta\over\sqrt{x^2+a^2-2ax\cos\theta}}$$

to the potential energy of the mass. Unlike the example of the spherical shell, this integral can't be done analytically. However for $x>\!>a$ you can make a Taylor expansion in $a/x$ of $(x^2+a^2-2ax\cos\theta)^{-1/2}$. If you keep all terms up to second order in $a/x$, and then integrate, you should get the result quoted.