How would you go about this when using partial fraction decomposition?

Question

$\frac{(u+1)^2}{(u^2+1)(u^2+3)}$ The solution of my professor says it's obvious that with
$\frac{A}{u^2+1} + \frac{B}{u^2+3}$

$A+B = 1 $

I'm curious why this is so obvious, any ideas?

Answer 1

$A+B$ is the coefficient of $u^{2}$ in the numerator.

Answer 2

$$\begin{align}\dfrac{A}{u^2+1} + \dfrac{B}{u^2+3} &= \dfrac{(u+1)^2}{(u^2+1)(u^2+3)} \\ \dfrac{\color{red}{A}u^2 + 3A +\color{red}{B}u^2 + B}{(u^2+1)(u^2+3)} &= \dfrac{\color{#2df}{u}^2+2u+1}{(u^2+1)(u^2+3)}\end{align}$$

Now if we compare the co - efficient of $u^2$ on both sides , we get : $\boxed{A+B=1}$.