Yikes, that's more complicated than I thought it would be.
Here's what I would suggest:
Divide the trapezoid into two triangles by drawing AC. Now instead of having all four sides pretend that you have the three sides and angle of your choice. Let's pretend you know b, c, d and angle B. Then you can solve triangle ABC. Drop a line down from B perpendicular to AD and you can solve the right triangle you just made. This gives you the height. Solve the corresponding right triangle on the other side of the trapezoid. At this point solving the rest of the trapezoid is simple.
Once you've got all those equations written down you should have enough equations to solve for any four unknowns. So you should be able to use those same equations to solve the trapezoid given the four side lengths.
Just a thought, sorry I'm too lazy to go through all the math. But anyway this method avoids having to use the expression they give for the height. (I don't like using expressions I can't derive.)