Intuition for why the difference between 2×2−xx2−x+1\frac{2x^2-x}{x^2-x+1} and x−2×2−x+1\frac{x-2}{x^2-x+1} is a constant?

Why is the difference between these two functions a constant?

$$f(x)=\frac{2x^2-x}{x^2-x+1}$$
$$g(x)=\frac{x-2}{x^2-x+1}$$

Since the denominators are equal and the numerators differ in degree I would never have thought the difference of these functions would be a constant.

Of course I can calculate it is true: the difference is $2$, but my intuition is still completely off here. So, who can provide some intuitive explanation of what is going on here? Perhaps using a graph of some kind that shows what’s special in this particular case?

Thanks!

---

BACKGROUND: The background of this question is that I tried to find this integral:

$$\int\frac{x dx}{(x^2-x+1)^2}$$

As a solution I found:

$$\frac{2}{3\sqrt{3}}\arctan\left(\frac{2x-1}{\sqrt{3}}\right)+\frac{2x^2-x}{3\left(x^2-x+1\right)}+C$$

Whereas my calculusbook gave as the solution:

$$\frac{2}{3\sqrt{3}}\arctan\left(\frac{2x-1}{\sqrt{3}}\right)+\frac{x-2}{3\left(x^2-x+1\right)}+C$$

I thought I made a mistake but as it turned out, their difference was constant, so both are valid solutions.

Answer

Attribution
Source : Link , Question Author : GambitSquared , Answer Author : Jared Goguen