This question tests your understanding of modal verbs, which are special verbs that express ability, permission, obligation, or possibility. Let's analyze the sentence:

"Sorry, I ______ go with you. I _____ finish my work."

The first blank requires a modal verb that expresses inability to perform an action (not going). The second blank requires a modal verb that expresses a strong obligation or necessity to perform an action (finishing work).

Step-by-Step Solution:

Step 1: Analyze the first blank. The speaker is apologizing for not being able to go. The correct modal to show inability is "can't".

Step 2: Analyze the second blank. The speaker provides the reason for the inability: they have an obligation to finish their work. The correct modal to show strong obligation or necessity is "must".

Final Answer: The correct option is can't/must.

The completed sentence reads: "Sorry, I can't go with you. I must finish my work." This makes logical and grammatical sense.

Why the other options are incorrect:

• couldn't/have to: "Couldn't" is the past tense of "can't" and is not appropriate for a present situation. While "have to" also shows obligation, the pair doesn't fit as well as "can't/must".
• mustn't/can: "Mustn't" means it is forbidden to do something, which doesn't fit the context of an apology. "Can" shows ability, which is the opposite of the required meaning of obligation.
• am/could: "Am" is a form of "to be" and is not a modal verb. It does not convey inability. "Could" shows possibility or past ability, not a present obligation.

Related Topic: Modal Verbs

Modal verbs are auxiliary (helping) verbs that modify the main verb to express concepts like permission, ability, obligation, and possibility. Common modal verbs include: can, could, may, might, must, shall, should, will, would.

Key Uses:

• Can/Could: Ability, possibility, permission.
• May/Might: Permission, possibility.
• Must/Have to: Obligation, necessity.
• Shall/Should: Suggestion, offer, advice.
• Will/Would: Future intention, conditional mood.

Related Formulae/Theory:

There is no mathematical formula, but a grammatical rule: Modal verbs are followed directly by the base form of the main verb (e.g., I must finish). They do not change form according to the subject (e.g., He can go, She can go).