Start by arranging both polynomials in decreasing powers of m (or n if you prefer; but I've chosen m). The problem becomes: divide by . Having done that, you will notice that m is always raised to a multiple of p, and n is always raised to a multiple of 2q. So it will save writing, and probably make the problem look easier, if we substitute x for , and y for . Then the problem becomes: divide by .
Next, you might notice that factorises as . So the problem will be solved if we can divide by 2x–y, and then divide the result by x–y. You should be able to do that by whichever method you have been taught (probably either synthetic division or long division of polynomials). That will give you the answer, when you finally replace x by and y by .