I don't understand partial fraction decomposition

[iScience]

if there is something like (x^2+3x+6) in the denominator for one of the terms in a partial fraction problem, why do we put Ax+B instead of just A? and if the denominator is (x^2+3x+6)^2, why do we do {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}? i was always just told to memorize it, but why do we set it up this way? even a link to a derivation will probably help. the only thing i can find online is the formulas, basically what i just wrote.. but i want to know why those setups are the way they are, ie the reason behind them.

 

[HallsofIvy]

Have you thought about the simplest cases? Suppose your initial fraction is just $(4x- 2)/(x^2+ 3x+ 6)$. Do you think you ought to be able to write that as $A/(x^2+ 3x+ 6)$? What would have happened to the "4x"?