Answer
$\dfrac{-1}{x}+\dfrac{4}{(x+1)} $
Work Step by Step
We will write the partial decomposition as:
$\dfrac{3x-1}{x(x+1)}=\dfrac{A}{x}+\dfrac{B}{x+1} ~~~~(a)$
This implies that $3x-1=A(x+1)+B x$
Plug $x=0$ to obtain: $A=-1$
Now, plug $x=-1$ to obtain: $B=4$
So, the equation (a) becomes:
$\dfrac{3x-1}{x(x+1)}=\dfrac{-1}{x}+\dfrac{4}{(x+1)} $
