Learn Hardcore German: Vor der Matheprüfung hatte sie Panik, deshalb nahm sie Nachhilfe bei einem Studenten. - Before the math exam she had panic, so she took tutoring with a student.

Exactly one element stands in the first position (here: Vor der Matheprüfung).
The conjugated verb must be in second position.
The subject and other elements come after that.

So the structure is:

Vor der Matheprüfung (position 1)
hatte (verb in position 2)
sie Panik (rest of the clause)

You could also say:

Sie hatte vor der Matheprüfung Panik.

Same meaning, just a different element in first position.

Why is it Panik haben instead of something like sie war panisch?

German often uses “to have + noun” where English uses “to be + adjective”.

Common expressions are:

Panik haben – to panic / to be in a panic
Angst haben – to be afraid
Hunger haben – to be hungry
Durst haben – to be thirsty

Sie hatte Panik is the most natural everyday way to say she panicked / she was in a panic.
You can say sie war panisch, but:

sie war panisch describes a general, lasting state or personality trait more than a short situation.
Panik haben / in Panik geraten fits better for a specific event like an exam.

What tense is hatte, and why not hatte … gehabt or hat … gehabt?

Hatte is the Präteritum (simple past) of haben.

Sie hatte Panik. → She panicked / she was in a panic.

In written narrative German, especially in stories, reports, and books, the Präteritum is normal for common verbs like sein, haben, werden, können, etc.

Sie hat Panik gehabt. → Perfekt (present perfect). Grammatically OK, but sounds more spoken / conversational.
Sie hatte Panik gehabt. → Plusquamperfekt (past perfect: she had had panic). You only use this if you explicitly need an earlier past compared to another past event. Here that is unnecessary and stylistically heavy.

So in a normal narrative sentence, hatte is the best choice.

Why is there a comma before deshalb?

The sentence contains two main clauses:

Vor der Matheprüfung hatte sie Panik,
deshalb nahm sie Nachhilfe bei einem Studenten.

In German, main clauses linked by a conjunctional adverb like deshalb, trotzdem, dann, jedoch are separated by a comma.

So the comma marks the boundary between two complete clauses, each with its own subject and verb.

How does deshalb affect the word order in the second clause?

Deshalb is a conjunctive adverb (therefore / because of that), not a subordinating conjunction.

In the second clause, German still uses the verb-second rule:

If deshalb comes first, it occupies position 1.
The verb must then come in second position.

So:

Deshalb (position 1)
nahm (verb, position 2)
sie Nachhilfe bei einem Studenten (rest of clause)

Alternative word orders:

Sie nahm deshalb Nachhilfe bei einem Studenten.
Sie nahm Nachhilfe bei einem Studenten, deshalb war sie beruhigter.

Both are fine; where you put deshalb can change emphasis, but the verb of the clause still stays in second position.

What exactly does Nachhilfe nehmen mean? Is Nachhilfe countable?

Nachhilfe means extra help with learning, usually private tutoring or remedial lessons outside normal school classes.

Nachhilfe nehmen = to take private lessons / get tutoring
Nachhilfe geben = to tutor / to give private lessons

About countability:

Nachhilfe itself is usually treated as uncountable (like help in English).
If you want to count the lessons, you use another noun, for example:
- Nachhilfestunde(n) – tutoring hour(s)
- Nachhilfestunden nehmen – to take tutoring hours / lessons

In everyday speech, people still say things like:

Sie nimmt Mathe-Nachhilfe. – She gets extra help in math.

but they don’t normally say eine Nachhilfe for a tutoring.

Why is it bei einem Studenten and not von einem Studenten or mit einem Studenten?

The verb Nachhilfe nehmen normally combines with bei:

Nachhilfe bei jemandem nehmen = to be tutored by someone / to take lessons from someone (at their place)

Prepositions here:

bei (+ dative): emphasizes “at / with (as a provider/host)”
- bei einem Studenten → the student is the person who gives the lessons.
von (+ dative): emphasizes origin or giver, more neutral: from
- Nachhilfe von einem Studenten is understandable, but less idiomatic.
mit (+ dative): means together with
- mit einem Studenten would sound like they are learning together with a student (as two learners), not that the student is the tutor.

So Nachhilfe bei einem Studenten nehmen is the standard, idiomatic phrase.

What case is einem Studenten, and how do you know?

Einem Studenten is dative singular masculine.

Reasons:

The preposition bei always takes the dative.
Student is a masculine noun: der Student.
The masculine indefinite article in dative singular is einem.
Student belongs to the n-declension (weak noun) group:
- nominative: der Student
- genitive: des Studenten
- dative: dem Studenten
- accusative: den Studenten

So with an indefinite article:

einen Studenten (accusative)
einem Studenten (dative)

Here bei forces dative → bei einem Studenten.

Could you also say Vor der Matheprüfung hatte sie Angst? What’s the difference between Angst and Panik?

Yes, Vor der Matheprüfung hatte sie Angst is perfectly correct.

Nuance:

Angst = fear, being afraid.
- Can be weak or strong, but neutral by default.
- Angst haben vor … is very common: Sie hatte Angst vor der Matheprüfung.
Panik = panic, a much stronger, more sudden, intense emotional reaction.
- Suggests losing control, maybe almost a panic attack.

So:

hatte sie Angst = she was afraid of the exam.
hatte sie Panik = she was really freaking out / in a panic about the exam.

Is Matheprüfung always written as one word? Can you say Mathe Prüfung?

In standard German, compound nouns are written as one word, so:

Matheprüfung = correct
Mathe Prüfung = incorrect

Structure:

Mathe (short for Mathematik)
Prüfung (exam)

Together: die Matheprüfung = the math exam / mathematics test.

You still capitalize the whole compound because nouns are capitalized in German, but you don’t put a space between the parts.