If I have 5 colored shapes shown that I can rotate, and I use each shape once, is it possible to place them so they fit perfectly in a rectangle?
Hint: Notice the checkerboard pattern of the rectangle, and consider how the shapes would be colored if they followed the same pattern.
Consider the rectangle colored like a checkerboard. Each tetromino covers exactly two white and two black squares except for the "T" shape which must cover either three white squares & one black square or three black squares & one white square. Since an equal number of black and white squares must be covered to include the complete checkerboard, it is impossible to fit the pieces in a 5 by 4 rectangle.