Why the Euclidean division of polynomials works? Why the Euclidean division of polynomials way actually works?
What is the deep insight behind it?
Thank you in advance.
 A: The key idea of the polynomial division algorithm is this: if the leading coefficient of the divisor $= 1$ (or is invertible), and the dividend has degree $\ge$ the divisor, then we can  $\rm\color{#c00}{scale}$ the divisor so that it has the same degree and leading coef as the dividend, then subtract it from the dividend, thereby canceling the leading term of the dividend; then recursively apply this process to the remaining part of the dividend, which has smaller degree (since we killed the leading term of the dividend), viz.
$$  (\overbrace{a x^{k+n} + f}^{\rm dividend}) - \color{#c00}{a x^k} (\overbrace{x^n + g}^{\rm divisor})\ =\ f-ax^kg$$
$$\ \Rightarrow\ \dfrac{a x^{k+n}+f}{x^n+g}\, =\ \color{#c00}{a x^k} +\!\!\! \underbrace{\dfrac{f-ax^k g}{x^n + g}}_{\large\rm recurse\ on\ this}$$
where the second equation arises from the first by dividing through by $\,x^n + g.\,$ The long division algorithm for polynomials is simply a convenient tabular arrangement of the process obtained by iterating this descent process till one reaches a dividend having smaller degree than the divisor.
Remark $\ $ If one seeks a deeper understanding one can view this as a special case of  multivariate polynomial division algorithms, such as the  Gröbner basis algorithm. One gains further insight from this more general perspective on the descent process, e.g. in terms of monomial orderings.
