Which rational numbers $\displaystyle t$ are such that
$\displaystyle 3t^3+10t^2-3t$
is an integer?
It would make things a bit clearer if you say something like, let $\displaystyle t=n/m$ be a rational number such that $\displaystyle 3t^3+10t^2-3t$ is an integer.
Now the cases $\displaystyle m=\pm 1$ correspond to $\displaystyle t$ being an integer, but $\displaystyle 3t^3+10t^2-3t$ is an integer when ever $\displaystyle t$ is an integer so all integer values of $\displaystyle t$ are trivially among those which make the expression in question an integer.
Or am I still misunderstanding something?
CB