<s>[INST] why does the quotient become messy when we switch the terms of the divisor when using long division? If we were to use long divison to divide $x^2 - y^2$ by $x + y$, the long division easily gives $x-y$. But if I were to use long division to divide $x^2 - y^2$ by $y + x$ instead, the quotient becomes lengthy and fruitless. Forgive me if I'm fundamentally flawed. I don't seem to understand this at all.
 [/INST] Dividing $x^2-y^2$ by $y+x$ gives:$$\frac{x^2}{y}-y-\frac{x^3}{y^2}+x+\frac{x^4}{y^3}-\frac{x^2}{y}-\frac{x^5}{y^4}+\frac{x^3}{y^2}+\frac{x^6}{y^5}-\frac{x^4}{y^3}-\frac{x^7}{y^6}+\frac{x^5}{y^4}+\frac{x^8}{y^7}-\frac{x^6}{y^5}-\cdots$$  The pattern of signs is + - - +, and every fractional term cancels with the fifth term following, leaving $-y + x$. So going at it this way is lengthy but not fruitless. It seems the order does not affect the result, only the ease of attaining it, at least in this example.  
</s>