Answer
$A = 1; B = 4$
Work Step by Step
The given equation is:
$$\dfrac{x+4}{x^2} =\dfrac{A}{x}+\dfrac{B}{x^2} ~~~(1)$$
On multiplying equation (1) with $x^2$, we obtain:
$$x+4= Ax+B ~~~~(2)$$
Now, we will put $x=0$ in order to eliminate $A$ from equation (2), we obtain:
$$B=4$$
Next, we will put $x=1$ in order to eliminate $B$ from equation (2), we obtain:
$$1 +4 = A+B \\ 5 =A+4 \\ A=1$$
Our required results are:
$$A = 1; B = 4$$