Learn Hardcore German: Sie hat das Angebot zuerst ablehnen müssen, nachher hat sie es dann angenommen. - She first had to decline the offer; afterwards she then accepted it.

With a modal verb plus another verb in the present perfect, German uses the “replacement infinitive” (Ersatzinfinitiv): both the main verb and the modal appear as infinitives at the end: … ablehnen müssen. You do not use the past participle abgelehnt there. So: Sie hat das Angebot … ablehnen müssen is correct; abgelehnt müssen is not.

Why isn’t it gemusst (e.g., hat … gemusst)?

When a modal has another verb with it, the modal also stays in the infinitive: … müssen. You use gemusst only when the modal stands alone: e.g., Das hat sie nicht gemusst (“She didn’t have to [do it]”). Here the modal governs ablehnen, so it must be müssen, not gemusst.

Could I say Sie musste das Angebot zuerst ablehnen instead?

Yes. That’s the simple past (Präteritum). In everyday German, both are fine: Sie musste … ablehnen (Präteritum) vs. Sie hat … ablehnen müssen (Perfekt). The choice is stylistic/regional; Perfekt is more typical in speech in many areas.

Why is the auxiliary haben used and not sein?

Modals take haben in the perfect, and transitive verbs like annehmen also take haben. Sein is mainly used with intransitive verbs of motion or change of state.

Why is zuerst placed where it is? Could it move?

Yes. Both are fine:

Sie hat das Angebot zuerst ablehnen müssen.
Sie hat zuerst das Angebot ablehnen müssen. Moving zuerst changes emphasis slightly (focus on time vs. object), but both are idiomatic.

Why do we have both nachher and dann? Isn’t that redundant?

It’s a bit redundant, but common in speech for emphasis and flow. You could say simply Nachher hat sie es angenommen or Dann hat sie es angenommen. Using both adds a “then afterwards” feel.

What’s the difference between nachher, danach, and später?

nachher: colloquial “later/afterwards.”
danach: neutral, a bit more written/formal “after that.”
später: “later” in general, not necessarily tied to a preceding event. All would work here: Nachher/Danach/Später hat sie es angenommen.

Why is it hat sie es dann angenommen and not hat sie dann es angenommen?

Pronoun objects (like es) generally come early in the middle field, before most adverbs. So … sie es dann … is the natural order. … sie dann es … sounds unidiomatic.

Could I drop es in the second clause?

No. German normally requires the object pronoun; you can’t omit it like English sometimes can. Nachher hat sie es angenommen is needed to refer back to das Angebot.

Are ablehnen and annehmen separable verbs? How does that affect the forms we see?

ablehnen is separable: present finite form splits (e.g., sie lehnt … ab), participle is abgelehnt. With a modal, it stays as the bare infinitive ablehnen at the end.
annehmen is separable; the participle is angenommen: Sie hat es angenommen.

Why is there no zu before ablehnen?

After modal verbs (dürfen, können, mögen, müssen, sollen, wollen), the following verb appears without zu: … ablehnen müssen, not … zu ablehnen müssen.

Is the comma between the two parts correct?

Yes. They’re two main clauses coordinated without a conjunction; German allows a comma here. A semicolon or a period would also be fine, depending on style.

Does Sie mean “she,” “they,” or polite “you” here?

It must mean “she.” The verb form hat is 3rd person singular. Polite Sie and plural sie would take haben.

Does … ablehnen müssen mean “must have declined” (inference) or “had to decline” (obligation)?

Obligation. For an inferential “must have,” German uses müssen with a past infinitive: Sie muss das Angebot zuerst abgelehnt haben (“She must have declined the offer at first”).

Could I use nachdem instead of nachher?

Yes, but the structure changes because nachdem introduces a subordinate clause with past perfect:

Nachdem sie das Angebot zuerst hatte ablehnen müssen, hat sie es angenommen. (Or: … ablehnen müssen hatte, both orders are seen.) This is more formal and emphasizes sequence.

Is … ablehnen müssen the only possible order of the two infinitives?

It’s the most common and recommended. You may also hear … müssen ablehnen, especially in some regions or registers, but stick with … ablehnen müssen as your default.

Can I front the object for emphasis?

Yes: Das Angebot hat sie zuerst ablehnen müssen; nachher hat sie es dann angenommen. Fronting das Angebot emphasizes that specific thing.