A clock seen through a mirror shows quarter past three. What is the correct time shown by the clock?

8 : 45

Quarter past three means the time shown in the mirror image is 3:15. However, the image is laterally inverted.

The hour hand of the mirror image is between 3 and 4. This means that the hour hand in the actual clock is between 8 and 9.
The minute hand of the mirror image is at 3 (15 minutes). This means that the minute hand in the actual clock is at 9(45 minutes).

So, the actual time will be 8:45.