smc

Factor the quadratic into $$\left(5x+\frac{12}{n}\right)\left(x+n\right)=0$$ where $-n$ is our integer solution. Then, $$k=\frac{12}{n}+5n,$$ which takes rational values between $-200$ and $200$ when $|n|\le 39$, excluding $n=0$. This leads to an answer of $2\cdot 39=\boxed{78}$.