2022 AMC 10A Problem 9

There are $5$ choices for the color of the bottom left rectangle. This forces there to be $4$ choices for the top left rectangle. The middle bottom rectangle touches both of the previous ones, so there are $3$ color options for this rectangle. The rectangle in the top right is also limited to $3$ colors since it touches the two previous rectangles. Finally, the rectangle in the bottom right also has $3$ color options. Multiplying these together, we get $5\cdot4\cdot3\cdot3\cdot3=540$ total colorings. Thus, D is the correct answer.