Answer
$A=-4$
Work Step by Step
With $f(0)=2$, substitute $2$ to $x$ and $2$ to $f(2)$ to obtain:
$f(x)=\dfrac{3x+8}{2x-A}$
$f(2)=\dfrac{3(0)+8}{2(0)-A}$
$2=\dfrac{8}{-A}$
Cross-multiply to obtain:
$-2(A)=8$
$-2A=8\\
A=\dfrac{8}{-2}\\
A=-4$