Nie mogę zawiązać sznurówek, bo moje ręce są bardzo zimne.

Because mogę already tells you the subject is I. It is the 1st person singular form of móc (to be able to / can).

So:

• mogę = I can / I’m able
• nie mogę = I can’t / I’m not able

Polish often drops subject pronouns when they are obvious from the verb ending. You can say Ja nie mogę..., but that adds emphasis, like I can’t... rather than someone else.

This is a very common question, because English can’t can mean two different things.

• nie mogę = I can’t / I’m unable to
  This usually refers to a situation, condition, or lack of possibility.
• nie umiem = I don’t know how to / I can’t because I never learned

In this sentence, the reason is cold hands, so the problem is temporary physical inability. That is why nie mogę is the natural choice.

Compare:

• Nie mogę zawiązać sznurówek, bo ręce mam zimne.
  = I can’t tie my shoelaces because my hands are cold.
• Nie umiem zawiązać sznurówek.
  = I don’t know how to tie shoelaces.

After móc (can / be able to), Polish normally uses the infinitive of the second verb.

So the pattern is:

• mogę zrobić = I can do
• mogę iść = I can go
• mogę zawiązać = I can tie

This is very similar to English, where can is also followed by a bare verb: can tie, can go, can do.

This is about aspect, which is very important in Polish verbs.

• zawiązać = perfective
  to tie something successfully / to get it tied
• zawiązywać = imperfective partner of zawiązać
  to be tying / to tie repeatedly / to tie in a general sense
• wiązać = a broader imperfective verb meaning to tie

In this sentence, the idea is: I can’t manage to get the shoelaces tied. That makes zawiązać very natural, because it focuses on the completed result.

Compare:

• Nie mogę zawiązać sznurówek.
  = I can’t get my shoelaces tied.

• Nie mogę zawiązywać sznurówek.
  = I can’t be tying shoelaces / I can’t do the activity of tying shoelaces.

  The second one is possible in some contexts, but it is less natural here.

Because after a negated verb, Polish often uses the genitive instead of the accusative for the direct object.

Positive sentence:

• Mogę zawiązać sznurówki.
  Here sznurówki is the normal object form.

Negative sentence:

• Nie mogę zawiązać sznurówek.
  Here the object changes to genitive plural: sznurówek.

So:

• sznurówki = nominative/accusative plural
• sznurówek = genitive plural

This is a classic Polish pattern often called the genitive of negation.

Yes. Ręka can cover both hand and arm depending on context. Polish does not divide these meanings in exactly the same way English does.

In this sentence, ręce clearly means hands, because hands are what you use to tie shoelaces.

If you specifically want to talk about the palms/hands, Polish also has dłonie, but zimne ręce is the normal everyday expression for cold hands.

Because they all refer to hands, which are plural.

Polish words agree with each other, so the possessive and adjective have to match the noun.

• moje = my for plural non-masculine-personal nouns
• ręce = hands
• są = are
• zimne = cold in the plural form matching ręce

You can compare:

• Moja ręka jest zimna.
  = My hand is cold.
• Moje ręce są zimne.
  = My hands are cold.

So the endings change because the noun changes from singular to plural.

You often can leave it out in Polish, especially with body parts, if it is obvious whose body you mean.

So this sentence is correct, but a very natural alternative would be:

• Nie mogę zawiązać sznurówek, bo mam bardzo zimne ręce.

That version sounds especially natural in everyday speech.

Using moje is not wrong. It can add a little emphasis or clarity. But Polish often avoids possessives with body parts more than English does.

Yes. Both can mean because, but they differ in style.

• bo = common, neutral, everyday
• ponieważ = more formal, more written

So in casual spoken Polish, bo is the most natural choice:

• Nie mogę zawiązać sznurówek, bo moje ręce są bardzo zimne.

A more formal version would be:

• Nie mogę zawiązać sznurówek, ponieważ moje ręce są bardzo zimne.

Both are correct.

Yes. Polish word order is more flexible than English word order.

The given version is a very natural neutral sentence:

• Nie mogę zawiązać sznurówek, bo moje ręce są bardzo zimne.

But you could also say:

• Bo moje ręce są bardzo zimne, nie mogę zawiązać sznurówek.

That puts more focus on the reason first.

You can also hear:

• Nie mogę zawiązać sznurówek, bo mam bardzo zimne ręce.

In Polish, changing the order usually changes emphasis more than basic meaning.

A few parts are especially worth noticing:

• nie sounds roughly like nye
• sz in sznurówek sounds like English sh
• ó sounds like u
• ą and ę are nasal vowels, but in normal speech their exact pronunciation changes depending on the sounds around them
• Polish stress is usually on the second-to-last syllable

So the main stress points are:

• MO-gę
• za-WIĄ-zać
• sznu-RÓ-wek
• BAR-dzo
• ZIM-ne

Also, don’t worry if the nasal vowels do not sound perfect right away. Learners often need time to get comfortable with ą and ę.