Answer
$\dfrac{-4}{x}+\dfrac{4}{x-1}$
Work Step by Step
We can notice from the given rational expression $\dfrac{A(x)}{B(x)}$ that $B(x)$ has only non-repeated linear factors.
So, we will write the given rational expression into partial fraction form as: $\displaystyle \frac{4}{x(x-1)}=\frac{A}{x}+\frac{B}{x-1} $
Now, write the RHS with a common denominator.
$ \displaystyle \frac{4}{x(x-1)}=\frac{A(x-1)+Bx}{x(x-1}\quad \quad(1)$
Next, equate the numerators to obtain:
$Ax-A+Bx=4
\\(Ax+Bx)-A=4
\\(A+B)x-A=4$
Equate the coefficients of the polynomials on the LHS and RHS to obtain:
$$-A =4 \implies A=-4$$
Also.
$$\\A+B=0 \quad \quad(2)$$
Plug $A=-4$ into the Equation $(2)$ to solve for $B$.
$$-4+B=0\implies B=4$$
Thus, the equation $(1)$ becomes:
$$ \displaystyle \frac{4}{x(x-1)}=\frac{-4}{x}+\frac{4}{x-1}$$
