Show That The Areas Of Triangle ABC And Triangle ACD Equal The Area Of Kite ABCD When BD = 18, ED = 12, And AE = 4?

You have not described the location of the point E, nor the symmetry of the kite. There is insufficient information to demonstrate what you want using numbers.

However, "by inspection" the sum of the areas of triangles ABC and ACD will be the area of kite ABCD. Segment AC divides the figure into those two named triangles.