How do I chose which factor goes under what constant?
I understand that $x^{2}+x = Ax+B$ or $Bx+C$ etc (That is not my question/concern)
My question is:
When I factor a quadratic, say for example:
$$\frac{x-1}{x^{2}+3x+2}$$
The denominator would factor out to $(x+2)(x+1)$ How do I know if A goes over $(x+2)$ or $(x+1)?$
Setting these up wrong seem to mess up my answer completely when I check on symbolab and emathhelp
Given
$$\frac{x-1}{x^2+3x+2}=\frac{x-1}{(x+2)(x+1)}$$
it doesn't matter if you set
$$\frac{x-1}{x^2+3x+2}=\frac{A}{x+2} + \frac{B}{x+1}\tag{1}$$ or $$\frac{x-1}{x^2+3x+2}=\frac{B}{x+2} + \frac{A}{x+1}\tag{2}$$
when you do partial fraction decomposition. In both cases, the constant for $A$ equals the constant for $B$ in the other case. Therefore, you will arrive at the same solution.
Both $A$ and $B$ refer to arbitrary constants. The important part is that you choose two different constants and then solve by the partial fractions decomposition.