Layton Raceway made easy with this simple drift boost trick.

Heres the trick

I see a lot of guides here telling players to simply press brake while turning, this is not completely correct.

You will have a much better time if you just use the handbrake button.
When coming into one of the turns, take the inner lighter coloured part of track. Now you need to do this: press 'X' and 'A' on your control around the same time while turning, this will pull the handbrake and make it a lot easier to get in to a power slide.
Once in the slide its all now up to you to pick the right line and keep it there. You will get it with practice.
But i can 100% tell you trying it with just the standard break method will likely get you no place fast.