The differential equation $\frac{d y}{d x}=\frac{x+y-1}{x+y+1}$ reduces to variable separable form by making the substitution

$x+y=v$

Let $x+y=v$. Then, $1+\frac{d y}{d x}=\frac{d v}{d x}$.

Substituting these values in the given differential equation, we get

$\frac{d v}{d x}-1=\frac{v-1}{v+1} \Rightarrow \frac{d v}{d x}=\frac{2 v}{v+1} \Rightarrow \frac{v+1}{2 v} d v=d x$

Clearly, it is in variable separable form.