The clock shows 15 minutes slow, which means the current time indicated is actually 15 minutes earlier than the actual time. Given the clock reads 3:45, the actual time is 3:45 + 15 minutes = 4:00. However, the actual time is also described as a quarter past an hour, which is a quarter past the previous hour. Therefore, the correct time is 4:15. But since we're told the clock initially showed 3:45 before it was adjusted to reveal the actual time, the time we're seeking must already consider this 15 minutes slow adjustment, making it 4:15. Thus, the correct time in HH:MM format is 04:15.