Try[f_,d_]:=Module[{},
Print[f/.m->n-1//Expand];
Print[f/.n->m+1//Expand];
Print[Table[f/d/.n->k/.m->k-1,{k,5}]];]

Try[m^6+m^3+2m^2+2m,6]

Try[n m(m^4-m^3+m^2+2),6]

Try[n m^5-n m(m^3-m^2+m-1)+n m(n-2)+2n m,6]

Try[m^6+m+2n m +n m (n-2),6]

Try[(n^2-n+1)n m(n^2-4n+5),6]

Try[n(n-1)((n-1)^4-(n-1)^3+(n-1)^2-(n-2)),1]

Try[2m^3-m,1]