Any polynomial whose coefficients are integers will, if it has an integer solution, have an integer solution that is a factor of the coefficient of the zeroth order term i.e. the constant. For example look at the general quartic:
where the coefficients
are integers. Well if there is an integer solution to this then it will be a factor of
. So look at that constant in your quartic and (assuming you have the right quartic) try each of it's factors to see if one is a solution. Then you can simply factor that solution out.