Sure! Here are the mistakes underlined and the corrected versions:

1. **Mistake:** the long
**Correct:** the longer
**Sentence:** The more complicated the puzzle gets, the longer it takes to solve.

2. **Mistake:** the most polluted
**Correct:** the more polluted
**Sentence:** The more polluted the air gets, the harder it is to breathe.

3. **Mistake:** get up
**Correct:** get down
**Sentence:** Despite the crowded streets, Tom decided to get down downtown by motorbike.

4. **Mistake:** came up with
**Correct:** came down with
**Sentence:** We had to cancel our trip because our dad came down with the flu.

5. **Mistake:** more inviting it looks
**Correct:** the more inviting it looks
**Sentence:** The cleaner the kitchen is, the more inviting it looks.

6. **Mistake:** The crowdeder
**Correct:** The crowdeder
**Sentence:** The more crowded the bus is, the longer the journey takes.

7. **Mistake:** hanging up
**Correct:** hanging out
**Sentence:** My brother enjoys hanging out with his colleagues at the local pub after work.

8. **Mistake:** the most luxurious
**Correct:** the more luxurious
**Sentence:** The more expensive the hotel gets, the more luxurious it becomes.

9. **Mistake:** carry in
**Correct:** carry out
**Sentence:** The committee is working together to carry out a project to organise a charity event.

10. **Mistake:** cut down out
**Correct:** cut down on
**Sentence:** We must cut down on water pollution to protect marine life and ecosystems.

11. **Mistake:** More complicated
**Correct:** The more complicated
**Sentence:** The more complicated the problem becomes, the more confused she feels.

12. **Mistake:** came down to
**Correct:** came down with
**Sentence:** They came down with food poisoning after eating at that restaurant.

Feel free to ask if you need further assistance!