The captain said that we ______ by next week.
a.finished
b.have finished
c.are to finish
d.had finish
Which answer is right and why?

Question

Answer

"are to finish". Present tense may have a value of future. The rest are all past tense, present perfect and past perfect. We could also say "had finish" if we were talking about a past plan that has now changed, but I think a wider context would be needed for that.

Answer

Here are several alternative ways to say this. There are subtle differences in meaning but they all mean approximately the same thing:

The captain said that 
...will be finished by next week
...we will finish by next week.
...will have finished by next week.
...we will be finishing by next week
...we will have been finished by next week.
...we are to finish by next week.
...we are finishing by next week.
...we would finish by next week.
...we would be finished by next week.

All your choices are grammatical except for (d).
Rather than asking yourself if a sentence is right or wrong, ask yourself instead
1. is it grammatical?
2. if it is grammatical, what does it mean?

Answer

Here ‘are to finish’ is similar to ‘must finish’ or ‘are required to finish’. The other answers are inconsistent with ‘by next week’ which is in the future.

Answer

Thanks for your help!