NOTE: Here, 'makes' is used to describe the action in present tense. 'Makes' is the third person form of the verb.
As the given sentence is in active voice, we will use the bare infinitive here.
Option B - The marker 'to' is used with the verb, when the sentence is in passive voice. As the given sentence is in active voice, we will use the verb in its original form without the marker 'to'. 'To do' can't be used here, as the verb 'do' is accompanied by the marker 'to'. Hence, option B is incorrect.
Option C - As the main verb 'makes' is already in third person singular form, the secondary verb should be in its original form. 'Does' is the third person singular form of the verb 'do'. Hence, option C is incorrect.
Option D - 'Doing' is the participle form of the verb 'do'. As explained above, we need the original form of the verb, hence 'doing' is the wrong option. Thus option D is incorrect.
Option A - 'Do' is a verb in its original form and is a bare infinitive. Hence, option A is the correct answer.
'The professor often makes the student do a lot of homework.'