Explanation:

The meaning of the word QUEST is a long or arduous search for something.

To find the synonym for the word QUEST we need the see the meaning of the given words

1. route - a way or course taken in getting from a starting point to a destination.

2. search - try to find something by looking or otherwise seeking carefully and thoroughly.

3. path - a way or track laid down for walking or made by continual treading.

4. destination - the place to which someone or something is going or being sent.

Hence after looking at the meanings it is clear that the most suitable synonym for the given word is 'search'.

Option 2 is the correct answer.