We divide the chessboard into 16 blocks of $2 \times 2$. There cannot be more than two red squares per block, otherwise there would be a red trimino. Therefore, there are at most 32 red squares. Conversely, it is immediate to find an example with 32 squares (place two
red squares diagonally in each block).

b) We use the same division and there are at least two red squares per block, so the answer is 32 red squares minimum.
On colors some cells of an $8 \times 8$ chessboard in red. How many cells can be colored at most if we want there to be no red tromino? How many cells can be colored at least if we want every tromino to have at least one red cell?