My friend and I were finding the series expansion of the function $\frac{2x}{x^2+1}$. I naturally went for the Maclaurin expansion method, using the differentials to find the series expansion. The answer I got (which is correct) is $$\sum_{n=0}^\infty (-1)^n 2x^{(2n+1)} $$ The problem is, this took me a while to differentiate enough to see the pattern. My friend, however did something much faster - he used long division! $$ \require{enclose} \begin{array}{r} \end{array} {x^2+1} \enclose{longdiv}{2x} $$ Apparently, instead of using the $x^2$ (the highest power of x) as the subject for his division, he used 1 (the lowest power of $x$), and he got the answer almost immediately. Example: 1 goes into $2x$ $2x$ times, and $(2x)({x^2+1}) = ({2x^3+2x})$. Subtract this from $2x$ and continue.
I couldn't find any evidence of this type of long division anywhere, and I'm having trouble understanding the logic behind it. How can he start subtracting something larger than $2x$ from $2x$? What sort of long division is this?
Could someone please help me understand?