Let's say we have an improper function:
$$P(x) = \frac{2x^3-3x^2+2x-4}{x^2+4}$$
If we do long division why does the polynomial can be expressed as "quotient + reminder/divisor"
$$P(x) =2x-3 +\frac{-6x+8}{x^2+4}$$
this does not look like Euclid's division algorithm
Let $P(x) = \frac{f(x)}{g(x)}$ with $f,g\in k[x]$ for some field $k$. By Euclidean division we have that there exist $q, r\in k[x]$ such that $$f(x) = q(x)g(x)+r(x),\ \deg r < \deg g \text{ or } r = 0.$$ Dividing by $g$ we get $$P(x) = q(x) + \frac{r(x)}{g(x)},\ \deg r < \deg g \text{ or } r = 0.$$
Note that this works the same as in case of numbers. For example, $11 = 2\cdot 5 + 1$, so $\frac{11}5 = 2 + \frac 15$.