<s>[INST] Decompose a fraction in a sum of two Let's say that I have this fraction:
$$ \frac{2x}{x^2+4x+3}$$
I would like to decompose in two fraction:
$$ \frac{A}{x+3} + \frac{B}{x+1}$$
Which is the procedure for that?  :)
 [/INST] $$x^2+4x+3=0$$
$$\Delta=4^2-4 \cdot 1 \cdot 3=4$$
$$x_{1,2}=\frac{-b \pm \sqrt{\Delta}}{2a}=\frac{-4 \pm 2}{2}=-3 \text{ or } -1$$
$$x^2+4x+3=(x+3)(x+1)$$
$$\frac{2x}{x^2+4x+3}=\frac{2x}{(x+3)(x+1)}=\frac{A}{x+3}+\frac{B}{x+1}=\frac{A(x+1)+B(x+3)}{(x+3)(x+1)}=\frac{(A+B)x+(A+3B)}{(x+3)(x+1)}$$
Since,$\frac{2x}{(x+3)(x+1)}=\frac{(A+B)x+(A+3B)}{(x+3)(x+1)}$,it must be:
$$A+B=2 \text{ and } A+3B=0$$
$$A+3B=0 \Rightarrow A=-3B$$
$$A+B=2 \Rightarrow -2B=2 \Rightarrow B=-1$$
$$A=3$$
Therefore:
$$\frac{2x}{x^2+4x+3}=\frac{3}{x+3}-\frac{1}{x+1}$$
</s>