Hi everyone!

The problem I need help with is:

Preform partial fraction decomposition: x / 4x^{2 }+ 4x + 1

In the solution to this exercise the author first factorizes the denominator, which factorizes into (2x + 1)^{2}, but than he writes that

x / (2x + 1)^{2}= A / (2x + 1) + B / (2x + 1)^{2}, I am unsure why this is the case? I understand that it can't be A / (2x + 1) + B / (2x + 1) either, but the authors approach seems wrong to me (don't worry I know I am wrong, hehe), because (2x + 1)^{2}is the original factorization of 4x^{2}+ 4x + 1, so adding (2x + 1) to it, would in my eyes give a bigger number than 4x^{2}+ 4x + 1 .

Also, my teacher in Calculus 1 at university said that for the exam it is most important that we learn how to draw graphs and than the rest will come on its own. I assume that was meant as a joke, but he never jokes, so now I am trying to read into his statement to see if there is a underlying hidden message in it.

Thanks to anyone for reading this and helping out