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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 $$ \int \frac{2x^4-x^3}{x^2+x-2} \,\text{d}x = \int (2x^2-x+5)\,\text{d}x + \int \frac{-7x+10}{x^2+x-2}\,\text{d}x$$

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    In fact it is precisely the (Euclidean) polynomial division algorithm, except it's performed explicitly in equational form, rather than having the equations be implicit in some tabular format. – 2017-01-15
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    It is sometimes also known as partial fraction decomposition. At least, I think this will give the same result. (I'm not sure if that final fraction needs more work.) – 2017-01-16
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    @TheGreatDuck No, that requires you to factor the denominator, which was never done. – 2017-01-16
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    @SimpleArt yes, but long division sometimes has the same effect. – 2017-01-16
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    @BillDubuque I had seen the implicit algorithm before, but it is much more enlightening to see it done this way. Are there any sources you could reccommend that explore this topic further? – 2017-03-08
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    @Ovi What precisely are you interested in exploring further? – 2017-03-08
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    @BillDubuque Sorry for the ambiguity, I am wondering if you have any nice sources about how various implicit algorithms have been derived, and which is the most efficient one. – 2017-03-08
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    @Ovi I doubt you will find that mentioned anywhere since the connection between the tabular form and the associated equations is obvious. It might be easier to find for the analog for the long division algorithm for integers, though that too is usually presented as a rote algorithm. If this is not clear then you can ask a question about it and someone will likely explain it at length. – 2017-03-08
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    @BillDubuque Okay thanks. – 2017-03-08

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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.