Sie braucht den Rückspiegel nicht neu einzustellen, denn gestern hat der Fahrlehrer alles geprüft.

It can mean either:

• she
• formal you (singular or plural)

Only context tells you which one is meant. German spells formal Sie with a capital S, but Sie at the beginning of a sentence is capitalized anyway, so here the form alone does not settle it.

Because Rückspiegel is the direct object of einstellen.

The base noun is:

• der Rückspiegel = the rear-view mirror

But in this sentence it is in the accusative case, so masculine der changes to den:

• nominative: der Rückspiegel
• accusative: den Rückspiegel

So den Rückspiegel means the rear-view mirror as the thing being adjusted.

It is a compound noun:

• Rück = back / rear
• Spiegel = mirror

So Rückspiegel literally means rear mirror or rear-view mirror.

German uses compound nouns very freely, so learners often need to get used to reading them as smaller parts joined together.

This is the pattern brauchen + nicht + zu-infinitive, which means to not need to do something.

So:

• Sie braucht den Rückspiegel nicht neu einzustellen
  = She/You does/do not need to readjust the rear-view mirror

A very important point: in this meaning, brauchen is not just ordinary need with a noun. It works almost like a modal expression:

• Ich brauche nicht zu warten = I do not need to wait
• Er braucht das nicht zu machen = He does not need to do that

Because einstellen is a separable verb.

Its parts are:

• ein- = prefix
• stellen = put / set / adjust

When a separable verb is used with zu in the infinitive, zu goes between the prefix and the verb stem:

• einstellen → einzustellen
• aufmachen → aufzumachen
• anzufangen from anfangen

So einzustellen is exactly the expected form.

Because nicht is negating the idea of needing to readjust.

The structure is:

• brauchen + object + nicht + zu-infinitive

So:

• Sie braucht den Rückspiegel nicht neu einzustellen

means:

• She does not need to readjust the mirror

The nicht belongs with braucht ... einzustellen, not with den Rückspiegel alone.

If you moved nicht, you could change the emphasis or make the sentence sound less natural.

Here neu does not mean new in the sense of a brand-new object. It means anew, again, or freshly.

So:

• neu einstellen = to adjust again / readjust / set anew

That is why the sentence is not saying the mirror is new. It is saying the mirror does not need another adjustment.

Yes, einstellen has several meanings depending on context. Common ones include:

• to adjust / set
• to hire
• to discontinue / stop

Here, because the object is den Rückspiegel, the meaning is clearly to adjust.

So:

• den Rückspiegel einstellen = to adjust the rear-view mirror

This is very common in German: one verb can have several meanings, and the object or context tells you which one is intended.

Both can often be translated as because, but they behave differently.

• denn is a coordinating conjunction
• weil is a subordinating conjunction

After denn, normal main-clause word order stays the same:

• ..., denn gestern hat der Fahrlehrer alles geprüft.

After weil, the finite verb goes to the end:

• ..., weil der Fahrlehrer gestern alles geprüft hat.

So the meaning is similar, but the grammar is different.

German main clauses follow the verb-second rule.

That means whatever comes first, the finite verb must come second.

Here the speaker puts gestern first for time emphasis:

• gestern = position 1
• hat = position 2
• der Fahrlehrer = after the verb

So:

• gestern hat der Fahrlehrer alles geprüft

If the subject came first, it would be:

• der Fahrlehrer hat gestern alles geprüft

Both are correct. The first version emphasizes yesterday more.

German often prefers the perfect tense in everyday language, especially in speech.

So instead of:

• gestern prüfte der Fahrlehrer alles

you often hear:

• gestern hat der Fahrlehrer alles geprüft

Both are grammatically correct, but the perfect is much more common in normal spoken German.

It means checked everything.

• alles = everything
• geprüft = checked / inspected / tested

So the idea is that the driving instructor already checked all the relevant things yesterday, which is why the mirror does not need to be readjusted now.

In this sentence, the comma before denn is required.

German uses a comma to separate the two main clauses:

• Sie braucht den Rückspiegel nicht neu einzustellen, denn gestern hat der Fahrlehrer alles geprüft.

So this is not just optional punctuation. It is standard German sentence structure.