Solution below.

Reveal solution by highlighting:

Imagine that we’re creating these rectangles by actually cutting a piece of paper. The first cut will produce 2 new pieces, and we have to choose one of these for the second cut.

The first cut in a 10×10 square can be one of the 9 horizontal lines or one of the 9 vertical lines. 9+9=18.

The next cut can be parallel to the first cut or not. If it’s parallel, there are 8 lines left to choose from. Then we have to divide by 2, because the order doesn’t matter. 18*8/2.

If the cuts aren’t parallel, I can make the second cut in either of the new pieces, along one of the 9 lines. In all 18*2*9.

18*8/2+18*2*9=396.