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.
Everything will be turned around, if you are dividing by $y+x$ then just turn the equation around to read $-y^2+x^2$ and it will come out nicely as you say. However, it does not matter the order.
$$\frac{x^2-y^2}{y+x}=-y+\frac{x^2+yx}{y+x}=-y+x$$


As you can see, you just have to be careful!:)
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.