Learn Hardcore Portuguese (Portugal): Nós queremos participar no projeto com o Pedro. - We want to participate in the project with Pedro.

Both mean the same thing. The verb form queremos already shows the subject is we, so in normal conversation people often omit nós, unless they want to stress we (as opposed to someone else).

What exactly is queremos? How is it formed from querer?

Queremos is the present indicative, 1st person plural of the verb querer (to want).

Present tense of querer in European Portuguese:

eu quero – I want
tu queres – you want (informal, singular)
ele / ela / você quer – he / she / you (formal) want
nós queremos – we want
vós quereis – you (plural, very rare in modern speech)
eles / elas / vocês querem – they / you (plural) want

So queremos simply means we want.

Why is participar in the infinitive? Could it be participamos instead?

After querer, Portuguese uses the infinitive form of the second verb:

querer + infinitive = to want to do something
- queremos participar = we want to participate
- quero aprender = I want to learn
- ela quer sair = she wants to go out

You cannot say:

✗ Nós queremos participamos no projeto…

That mixes two conjugated verbs incorrectly. You must keep the second verb in the infinitive:

✓ Nós queremos participar no projeto…

What does no mean in participar no projeto? Is it a contraction?

Yes. No is a contraction of:

em (in / on / at)
- o (the, masculine singular)

So:

em + o = no
participar em o projeto → participar no projeto

Literally: participate in the project.

Why participar no projeto and not participar do projeto?

In European Portuguese, the normal pattern is:

participar em something
- participar em reuniões – to participate in meetings
- participar em projetos – to take part in projects
- participar no projeto – participate in the project (em + o → no)

Participar de is much more common in Brazilian Portuguese:

participar do projeto (BR) = participar no projeto / participar do projeto (PT, less usual)

So for standard European Portuguese, participar no projeto is the most natural choice.

Why is there an article before the name: com o Pedro and not just com Pedro?

In European Portuguese, it is very common to use a definite article before personal names in everyday speech:

o Pedro, a Ana, o João, a Marta

So:

com o Pedro = with Pedro

You can say com Pedro, and it is correct, but:

with article (com o Pedro) → very common, sounds more natural in casual European Portuguese
without article (com Pedro) → a bit more formal or neutral; also more typical in Brazilian Portuguese

Meaning-wise, both mean the same thing here.

Does com o Pedro mean the project belongs to Pedro, or just that Pedro is included?

In this sentence, com o Pedro simply adds Pedro as a participant:

Nós queremos participar no projeto com o Pedro.
→ We want to take part in the project together with Pedro.

If you wanted to say the project belongs to Pedro, you’d say something like:

o projeto do Pedro – Pedro’s project
participar no projeto do Pedro – to participate in Pedro’s project

So com o Pedro is about who is together with us, not about ownership of the project.

Is the word order fixed, or could I say Nós queremos com o Pedro participar no projeto?

That alternative order sounds unnatural in Portuguese. The normal order is:

Subject: Nós
Conjugated verb: queremos
Infinitive verb: participar
Prepositional phrase: no projeto
Extra phrase: com o Pedro

So:

✓ Nós queremos participar no projeto com o Pedro.

You can move nós or drop it, but you generally keep the two verbs together:

Nós queremos participar no projeto com o Pedro.
Queremos participar no projeto com o Pedro.

Putting com o Pedro between queremos and participar is grammatically possible but sounds awkward:

? Nós queremos com o Pedro participar no projeto. → technically understandable, but not natural in everyday speech.

How would this sentence look in Brazilian Portuguese?

A natural Brazilian version would be:

Nós queremos participar do projeto com o Pedro.
or simply
Queremos participar do projeto com o Pedro.

Key differences:

In Brazil, participar de is standard:
- participar do projeto (de + o) instead of participar no projeto.
Brazilians usually don’t use the definite article with names in this context, but com o Pedro is still very possible.
- com Pedro and com o Pedro both work; com Pedro may sound a bit more typical.

So the main systematic difference here is no projeto (PT) vs do projeto (BR).

How do I pronounce Nós queremos participar no projeto com o Pedro in European Portuguese?

An approximate phonetic guide (European Portuguese):

Nós → /nɔʃ/ (like “nosh” with a closed o)
queremos → /kəˈɾemuʃ/ (the first e is reduced, like a quick “uh”; final s like “sh”)
participar → /pɐɾtiʃiˈpaɾ/ (first a reduced to /ɐ/; r at the start of a syllable is a light tap)
no → /nu/
projeto → /pɾuˈʒɛtu/ (the j is like the s in “measure”)
com → /kõ/ (nasal vowel, like French “bon”)
o → /u/ (very short, like “oo”)
Pedro → /ˈpedɾu/

Said fluently, many vowels are reduced and words link together:

Nós queremos participar no projeto com o Pedro.
→ something like: /nɔʃ kəˈɾemuʃ pɐɾtiʃiˈpaɾ nu pɾuˈʒɛtu kõ u ˈpedɾu/

Is there any difference in meaning between Nós queremos participar… and something like Nós gostaríamos de participar…?

Yes, there is a nuance:

Nós queremos participar no projeto com o Pedro.
→ Direct, neutral: We want to participate in the project with Pedro.
Nós gostaríamos de participar no projeto com o Pedro.
→ More polite / softer: We would like to participate in the project with Pedro.

So queremos is a straightforward statement of desire; gostaríamos de is more tentative and polite, often used in formal requests.

What gender is projeto, and how does that affect the sentence?

Projeto is masculine singular:

o projeto – the project
um projeto – a project
este projeto – this project

That’s why:

em + o projeto → no projeto
com o projeto (if you ever needed that form)
Any adjectives agreeing with it would be masculine singular:
- um projeto interessante – an interesting project
- o projeto novo – the new project

In the sentence Nós queremos participar no projeto com o Pedro, the masculine gender of projeto is reflected in the article o inside no.