Given a multivariate polynomial $f(x_1,\dots,x_n)$ with integer coefficients, how to find an integer $m$ (if it exists) such that $f(x_1,\dots,x_n) + m$ factors into polynomials of smaller degrees?

Are there any simple criteria to identify cases when such $m$ does not exist?

Is it possible that more than one suitable values of $m$ exist?