<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] $$ \frac{2x}{x^2+4x+3}=\frac{2x}{x^2+x+3x+3}=\frac{2x}{x(x+1)+3(x+1)}=$$
$$=\frac{2x}{(x+3)(x+1)}=\frac{A}{x+3}+\frac{B}{x+1}$$
$$2x=Ax+A+Bx+3B=(A+B)x+A+3B$$
You needs to solve the system
$$A+B=2,A+3B=0$$
</s>