Answer: 870

Small Hint:

The first equation implies that $b-a=d-c$

Big Hint:

Set $x=b-a=d-c$ and factor $bc-ad$ in terms of $x$ and $c-a$

Solution:

Let $x=b-a=d-c>0$ and $y=c-a.$ Then $$\begin{aligned}bc-ad&=(a+x)c-a(c+x)\\&=x(c-a)=xy=93.\end{aligned}$$ Since $b<c,$ we need $x<y.$ The positive factor pairs are $(x,y)=(1,93)$ and $(3,31).$ For the first, $d=a+94<500$ gives $405$ choices for $a.$ For the second, $d=a+34<500$ gives $465$ choices. The total is $405+465=870.$