In order to fill in the blank with the correct auxiliary verb, let's understand what auxiliary verbs are. Auxiliary verbs are used to form tenses, moods and forms of other verbs. For example - I am a woman. Here, 'am' is an auxiliary verb as it indicates that the sentence is in simple present tense.
NOTE: 'The boys' is a plural subject and hence we need the plural form of the auxiliary verb. The verb 'coming' indicates that the sentence is in the continuous tense.
Option B - 'Is' is the singular form of the auxiliary verb used with a singular subject. As we need the plural form of the auxiliary verb, we can't use 'is' here. Hence, option B is incorrect.
Option C - 'Have' is the plural form of the auxiliary verb used with a plural subject. 'Have' is often used in the simple present tense to show possession or ownership. 'Have coming' is grammatically incorrect. Also, the sentence is not about possession, that's why we can't use 'have' here. Hence, option C is incorrect.
Option D - 'Has' is the singular form of the auxiliary verb which is used with a singular subject. As the sentence requires the plural form of the auxiliary verb, we can't use 'has' here.
Option B - 'Are' is the plural form of the auxiliary verb used with the plural subject. It is often accompanied by the participle form of the verb to make the present continuous tense. As the sentence has the verb in the participle form, we can use 'are' with it. Hence, option B is the correct answer.
'The boys, as well as their mothers, are coming.'