smc

For $x^2+x-n$ to factor into a product of two linear factors, we must have $x^2+x-n=(x+a)(x+b)$, where $a$ and $b$ are integers. By expansion of the product of the linear factors and comparison to the original quadratic, $ab=-n$ $a+b=1$. The only way for this to work if $n$ is a positive integer is if $a=-b+1$. Here are the possible pairs: $a=-1,b=2$ $a=-2,b=3$ $⋮$ $a=-9,b=10$ This gives us 9 integers for $n$, $\boxed{9}$.