What is the length of AD or AC? We need the length of some sides to determine what the area is. Many trapezoids that fit your description could have different side lengths, leading to different areas.
I cannot help you too much because insufficient information was given, but I can tell you that ABCD is an isosceles trapezoid. Angle ADC = Angle BCD, Angle DAB = Angle CBA. Since AC bisects angle DAB, DB also bisects angle ABC.
"The intersection of AC and DB is Point O.
Since the top triangle (ABO) is 4 times smaller than the bottom triangle (DOC), and the triangles on the side are the same area. "
What are you talking about PM ? This makes no sense.
What does this mean [ABCD] = 42 ??
I have never seen that notation before.