<s>[INST] What's the name of the following method for dividing polynomials? It's not long-division nor synthetic division I saw this method in some random PDF and am intrigued of the exact method used. I can't find any page of this method on the web because I'm not sure what you'd call this method.
Here is the method:

For solving
  $$\int \frac{2x^4 + x^3}{x^2 + x - 2} \,\text{d}x$$
  Observe that
  $$\begin{align*}
2x^4 + x^3
&= 2x^2 (x^2+x-2) - x^3+4x^2 \\
&= 2x^2 (x^2+x-2) - x(x^2+x-2) + 5x^2-2x \\
&= 2x^2 (x^2+x-2) - x(x^2+x-2) + 5(x^2+x-2) - 7x+10 \\
&= (2x^2-x+5)(x^2+x-2) - 7x+10
\end{align*}$$
  and then
  $$
\int \frac{2x^4-x^3}{x^2+x-2} \,\text{d}x
= \int (2x^2-x+5)\,\text{d}x
+ \int \frac{-7x+10}{x^2+x-2}\,\text{d}x$$

 [/INST] What you have is literally polynomial long division written out without division signs.  Indeed, what is written out here is the essence of polynomial long division, which is all about finding the coefficient of the factor that returns the highest degree term in the original polynomial.
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