a) You haven't shown that there are no solutions where exactly one of b and d is 0.  This is easy enough to do.  For instance, if d = 0 and b is non-zero, then c = 1 and a = a +1, so there is no solution.

b) At one point you start out assuming that b and d have the same sign, calculate that bd = -1, and reach a solution that violates your assumption.  Upon getting to bd = -1, you should just conclude that there are no solutions where b and d have the same sign.