d. A lot of
Explanation:
Does the teacher give you homework?
\/
Does the teacher give you a lot homework?
a) Many is to do with a certain number of something when asking. How many cars? How many potatoes? Etc.
b) Much is similar to many. How much do you want? How much is that? Etc.
c) A few does not make sense here. A few is more used when answering, not asking a question.
d) A lot of makes sense and completes this sentence.
Have a nice day!
I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)
- Heather