jmc

Let $\sqrt{x+150}=n$, where $n$ is a positive integer. Then $x+150=n^2$, so $x=n^2-150$. We see that $x$ is negative for $n=1$, 2, 3, $\dots$, 12, but positive for $n\ge 13$, so the number of possible values of $x$ is $\boxed{12}$.