Question 903823
Not remembering any instruction during Introductory Algebra from so long ago that dealt with factoring degree 4 trinomial, eventually further instruction combined from Introductory and College Algebra contained enough knowledge to use polynomial division for trying to handle something like you have; including Rational Roots Theorem, although you are not interested in roots here.


There is m^4 and n^4, the n^4 term having coefficient 72.  Factors of 72 are 2, 3, 6, 12, 24,36.


You would want to test DIVISORS of {{{m-2n}}}, {{{m+2n}}}, {{{m-3n}}}, {{{m+3n}}}, {{{m-6n}}}, {{{m+6n}}}, ... possibly others.   Your DIVIDEND must be formed as {{{m^4+0m^3*n+(-38)m^2n^2+0mn^3+72n^4}}};


After finding the quotient with remainder of zero, you next use whatever skills you have for factoring, or you might try another polynomial division on the quotient, choosing a divisor that you believe is worth testing.