Why is it guaranteed that all rational functions $\frac{R(x)}{Q(x)}$, where the degree of $R$ is less than $Q$, can be written as the sum of partial fractions of the form $\frac{Ax + B}{(ax^2+bx+c)^i}$? For example, the simple case $$ \frac{P(x)}{(x+a)(x+b)}=\frac{A}{x+a}+\frac{B}{x+b}\\ \implies P(x) = A(x+b) + B(x+a) $$ This example reminds me of the concept of linear independence from Linear Algebra, perhaps this statement is true because $(x+b)$ and $(x+a)$ are not multiples of each other, and therefore some linear combination of them has to equal $P(x)$.
Stewart's Calculus only says that "a theorem in algebra guarantees that it is always possible to do this." I've tried reading the proofs found on Wikipedia and from this previous post but I'm having a hard time understanding them.