Muszę dokładnie sprawdzić formularz, żeby nie popełnić błędu.

Muszę is the 1st person singular form of musieć: to have to / must.

So muszę sprawdzić means I have to check or I must check. In many everyday contexts, Polish muszę can cover both English ideas, and the difference between must and have to is usually not important.

After muszę, Polish normally uses an infinitive:

• muszę sprawdzić = I have to check
• musisz sprawdzić = you have to check
• musi sprawdzić = he/she/it has to check

Because sprawdzić is a perfective verb. It presents the action as a complete, finished task: checking the form all the way through.

That fits this sentence well: the speaker needs to complete the check in order to avoid an error.

Compare:

• sprawdzić = check, verify, finish checking once
• sprawdzać = be checking, check repeatedly, check as an ongoing/habitual activity

So:

• Muszę sprawdzić formularz = I need to check the form (one complete check)
• Muszę sprawdzać formularz would suggest something more repeated or ongoing, which is not the most natural meaning here.

Here dokładnie means carefully, thoroughly, or in detail.

In other contexts, dokładnie can also mean exactly, but in this sentence it describes how the form should be checked:

• dokładnie sprawdzić formularz = check the form carefully/thoroughly

So it is not exactly in the English sense here.

Formularz is the direct object of sprawdzić, so grammatically it is in the accusative case.

However, formularz is a masculine inanimate noun, and for many masculine inanimate nouns, the accusative singular looks the same as the nominative singular.

So:

• nominative: formularz
• accusative: formularz

The case has changed in function, but not in visible form.

Here żeby means so that or in order to. It introduces a purpose clause.

So:

• Muszę dokładnie sprawdzić formularz, żeby nie popełnić błędu. means:
• I have to check the form carefully so that I don't make a mistake.

The second part explains the purpose of the first part.

Yes. Aby is a more formal or literary alternative to żeby.

So you can say:

• Muszę dokładnie sprawdzić formularz, aby nie popełnić błędu.

That means the same thing, but sounds a bit more formal. In everyday spoken Polish, żeby is usually more common.

Polish often uses żeby + infinitive when the subject of both actions is the same.

Here, the same person:

• must check the form and
• wants not to make a mistake

So żeby nie popełnić błędu is a natural, compact structure.

A fully conjugated version is also possible, for example:

• żebym nie popełnił / popełniła błędu

But the infinitive version is simpler and very natural in this kind of sentence.

Yes. Popełnić błąd is a very common and natural Polish expression.

Even though English says make a mistake, Polish uses a verb closer to commit:

• popełnić błąd = make a mistake / commit an error

So it is worth learning as a set phrase.

Related forms:

• popełnić błąd = to make a mistake
• popełniać błędy = to make mistakes, to keep making mistakes

Because of negation.

Without negation, the object is normally in the accusative:

• popełnić błąd = to make a mistake

With nie, Polish very often changes the object to the genitive. This is called the genitive of negation:

• nie popełnić błędu = not to make a mistake

So:

• błąd = accusative/nominative form
• błędu = genitive singular

In this sentence, błędu is the standard and expected form.

Because żeby introduces a subordinate clause, and in Polish subordinate clauses are normally separated by a comma.

So the comma in:

• Muszę dokładnie sprawdzić formularz, żeby nie popełnić błędu. is standard punctuation.

Yes. Polish word order is more flexible than English word order, because case endings carry a lot of the grammatical information.

The original sentence has a very neutral, natural order:

• Muszę dokładnie sprawdzić formularz, żeby nie popełnić błędu.

But other orders are possible for emphasis, for example:

• Formularz muszę dokładnie sprawdzić, żeby nie popełnić błędu.

That version emphasizes formularz more strongly. For a learner, the original version is the safest default.