Je suis obligée de rentrer tôt, mais je peux passer à la boulangerie après avoir acheté les poireaux.

Because the speaker is female.

In je suis obligé(e) de..., the adjective obligé / obligée agrees with the person speaking:

• Je suis obligé de... = said by a man
• Je suis obligée de... = said by a woman

So the extra -e shows feminine agreement.

It means to have to, to be required to, or to be forced/obliged to do something.

So:

• Je suis obligée de rentrer tôt means I have to go home early

This structure is very common in French:

• Il est obligé de partir
• Nous sommes obligés d’attendre

It often sounds a little stronger than simply using devoir, because it suggests some necessity or constraint.

Because obligé(e) is normally followed by de + infinitive.

So the pattern is:

• être obligé(e) de faire quelque chose

Examples:

• Je suis obligée de travailler
• Tu es obligé de venir
• Elle est obligée d’attendre

This is just the standard construction French uses after this adjective.

Because after de in the pattern être obligé(e) de + verb, the next verb stays in the infinitive.

So:

• Je suis obligée de rentrer
• not Je suis obligée je rentre

French often uses one conjugated verb followed by an infinitive:

• je peux passer
• je vais partir
• je dois travailler
• je suis obligée de rentrer

All three can involve the idea of going back, but they are not used in exactly the same way.

• rentrer often means to go back home or to return inside
• revenir means to come back
• retourner means to go back / return to a place

In this sentence, rentrer tôt most naturally means go home early.

So Je suis obligée de rentrer tôt sounds very natural for someone saying they must get home early.

Because tôt is an adverb, and in French adverbs often come after the infinitive they modify.

So:

• rentrer tôt = to go home early

This is normal word order in French. Compare:

• partir vite
• arriver tard
• se lever tôt

English often puts the adverb in a similar place, but French learners still need to get used to how naturally it comes after the verb here.

Because pouvoir is followed directly by an infinitive, with no de.

So:

• je peux passer
• tu peux venir
• nous pouvons attendre

Some French expressions need de before the infinitive, but modal verbs like pouvoir, devoir, and vouloir do not.

Compare:

• Je suis obligée de rentrer
• Je peux passer

So the de belongs to obligée, not to peux.

Here passer à + place means to stop by or to drop by a place.

So:

• je peux passer à la boulangerie = I can stop by the bakery

This is an idiomatic use of passer. It does not mean simply to pass in the English sense.

Other examples:

• Je vais passer à la pharmacie
• Tu peux passer au supermarché ?
• Je passe chez Paul ce soir

Because French commonly uses à with shops and businesses to mean at / to that place.

So:

• à la boulangerie = to the bakery / at the bakery
• à la pharmacie
• au supermarché
• à la banque

This is a very common pattern with places you go to for a service or purchase.

Because when French wants to say after doing something, it usually uses après + infinitive perfect:

• après avoir + past participle
• or with some verbs, après être + past participle

So:

• après avoir acheté les poireaux = after buying the leeks

This structure shows that the buying happens before the stopping by the bakery.

More examples:

• Après avoir mangé, je pars
• Après être arrivé, il a appelé
• Après avoir fini, nous sommes sortis

Using après acheter would sound wrong in standard French.

Because acheter normally forms its compound tenses with avoir.

The structure here is basically the infinitive version of a past action:

• avoir acheté = to have bought

Just like:

• J’ai acheté in a normal finite tense
• après avoir acheté in this infinitive construction

Only certain verbs use être instead, such as verbs of movement or reflexive verbs:

• après être arrivé
• après s’être levé

Because in avoir acheté, the past participle usually does not agree unless there is a preceding direct object.

Here, les poireaux comes after the verb, so there is no agreement:

• avoir acheté les poireaux

So acheté stays in its basic form.

If there were a preceding direct object, agreement could happen:

• les poireaux que j’ai achetés

But that is a different structure.

Both can be possible in different contexts, but les poireaux suggests specific leeks that are already identified in the situation.

For example, maybe the speaker has already mentioned them, or both people know which leeks are meant.

• acheter les poireaux = buy the leeks
• acheter des poireaux = buy some leeks

So les makes the noun more definite or specific.

Because mais connects two clauses, and a comma before it is very normal in French writing.

The sentence has two main ideas:

• Je suis obligée de rentrer tôt
• mais je peux passer à la boulangerie...

The comma helps separate those two parts clearly. It works much like English punctuation before but.

Yes. Je peux is the normal everyday form.

• je peux = standard, common spoken and written French
• je puis = much more formal, less common, often found in inversion such as puis-je... ?

So in this sentence, je peux passer is exactly what most people would say.

Only obligée would change:

• Je suis obligé de rentrer tôt, mais je peux passer à la boulangerie après avoir acheté les poireaux.

Everything else stays the same.

That is because the feminine marking only affects the adjective agreeing with the speaker.