I was helping a high-school student last night whose teacher had given as a homework problem the division $$\frac{15x^4-y^2}{x^2+y};$$ I tried a heuristic involving splitting off a difference of squares to end up with $$15x^2-15y+\frac{14y^2}{x^2+y},$$ but I was not satisfied because the remainder has the same degree as the denominator and normally problems like these should end up with a denominator of lower degree.
I next tried variants of synthetic division "with respect to $y$" and "with respect to $x^2$" and got nothing simpler.
I then tried the method outlined in Karl's Calculus Tutor to perform long division, first with the original fraction and then with the remainder term arrived at with the first method that came to my head, and I kept looping; unlike numerical or single-variable long division, multi-variable long division doesn't seem to follow an ordered progression inexorably leading toward a definite remainder of lower degree (in the case of numerical long division, of smaller absolute value) than the denominator, except in simple cases like in the example from Karl's Calculus Tutor.
Best Answer
See Chapter 2, Section 3 (p. 61) in the book Ideals, varieties, and algorithms by Cox, Little & O'Shea. (Google books link.)