Learn Hardcore German: Eine sorgfältige Korrektur der Hausarbeit zeigt, wie ernst du die Gruppenarbeit nimmst. - A careful correction of the term paper shows how seriously you take the group work.

the feminine nominative indefinite article eine
plus the correct adjective ending for feminine nominative with an indefinite article, which is -e → sorgfältige

Pattern:

eine sorgfältige Korrektur (fem., nom.)
ein sorgfältiger Lehrer (masc., nom.)
ein sorgfältiges Kind (neut., nom.)

What case is der Hausarbeit, and why is that form used here?

Der Hausarbeit is in the genitive case, singular, feminine.

Structure:

Eine sorgfältige Korrektur der Hausarbeit
→ Korrektur is the main noun.
→ der Hausarbeit depends on Korrektur and answers “correction of what?”.

Some nouns commonly take a genitive complement in German, and Korrektur is one of them:

die Korrektur des Textes – the correction of the text
die Korrektur der Fehler – the correction of the mistakes
die Korrektur der Hausarbeit – the correction of the term paper / homework

Because Hausarbeit is feminine (die Hausarbeit), its genitive singular form is der Hausarbeit.

So:

Nominative: die Hausarbeit
Genitive: der Hausarbeit
Dative: der Hausarbeit
Accusative: die Hausarbeit

Why is ernst in wie ernst du die Gruppenarbeit nimmst not marked with an ending like ernste or ernsten?

Here ernst is an adverb-like predicative adjective, not an adjective directly modifying a noun.

Compare:

eine ernste Gruppenarbeit → adjective before a noun, takes an ending
du nimmst die Gruppenarbeit ernst → ernst describes how you take it, and does not take an ending

In wie ernst du die Gruppenarbeit nimmst, the focus is on the degree of seriousness. Ernst belongs conceptually to the verb phrase ernst nehmen (to take seriously). It behaves like many German adjectives when used predicatively or adverbially:

Er arbeitet schnell. – He works fast. (no ending)
Die Aufgabe ist schwer. – The task is difficult. (no ending)
Du nimmst die Gruppenarbeit ernst. – You take the group work seriously. (no ending)

So you don’t say du nimmst die Gruppenarbeit ernste/ernsten – that would be ungrammatical here.

What exactly does ernst nehmen mean, and is it a fixed expression?

Yes, ernst nehmen is a very common, semi-fixed expression meaning “to take (something) seriously.”

Pattern:

etwas ernst nehmen – to take something seriously
- Ich nehme die Prüfung ernst. – I take the exam seriously.
- Sie nimmt ihre Gesundheit sehr ernst. – She takes her health very seriously.

In your sentence:

wie ernst du die Gruppenarbeit nimmst
→ literally: how seriously you take the group work

Grammatically, nehmen is the main verb, and ernst behaves like an adverb, but from a learner’s point of view, it’s useful to learn the whole chunk etwas ernst nehmen as a pattern.

Why does the verb nimmst go to the end in wie ernst du die Gruppenarbeit nimmst?

Because wie ernst du die Gruppenarbeit nimmst is a subordinate clause embedded inside the main clause.

Main clause: Eine sorgfältige Korrektur der Hausarbeit zeigt ...
→ verb in second position: zeigt
Subordinate clause: ... wie ernst du die Gruppenarbeit nimmst.
→ introduced by the subordinator wie
→ in German subordinate clauses, the conjugated verb goes to the very end.

Typical pattern:

..., dass du die Gruppenarbeit ernst nimmst.
..., weil du die Gruppenarbeit ernst nimmst.
..., wie ernst du die Gruppenarbeit nimmst.

So the word order rule is:
subordinator + (other elements) + subject + ... + conjugated verb at the end

Could you also say zeigt, dass du die Gruppenarbeit ernst nimmst? What is the difference from zeigt, wie ernst du die Gruppenarbeit nimmst?

Yes, both are correct, but they differ in nuance.

... zeigt, dass du die Gruppenarbeit ernst nimmst.
- Focus: the fact that you take it seriously (yes vs. no).
- Rough meaning: ... shows that you take the group work seriously.
... zeigt, wie ernst du die Gruppenarbeit nimmst.
- Focus: the degree or extent of seriousness (how much).
- Rough meaning: ... shows how seriously you take the group work.

So:

If the main idea is “You do (indeed) take it seriously”, use dass.
If the main idea is “It shows the level of seriousness”, use wie ernst.

In English we often say “how seriously”, and German mirrors that with wie ernst.

What is the difference between Hausarbeit and Hausaufgabe? Both can mean “homework,” right?

They overlap in meaning in everyday speech, but there is an important difference:

die Hausaufgabe (Pl: die Hausaufgaben)
- normal school homework / exercises given for home
- common in school context
- Ich habe viele Hausaufgaben. – I have a lot of homework.
die Hausarbeit (Pl: die Hausarbeiten)
1. At university / college:
  - a written paper, term paper, seminar paper, coursework
  - usually longer and more formal
  - Ich muss noch meine Hausarbeit für das Seminar schreiben.
2. In everyday life:
  - housework (cleaning, cooking, etc.)
  - Er hilft nie bei der Hausarbeit. – He never helps with the housework.

In your sentence, context strongly suggests the academic/term paper meaning:
Eine sorgfältige Korrektur der Hausarbeit → A careful correction of the term paper

What are the genders and plurals of Hausarbeit and Gruppenarbeit, and which articles do they use?

Both are feminine nouns.

Hausarbeit
- singular: die Hausarbeit
- plural: die Hausarbeiten
Gruppenarbeit
- singular: die Gruppenarbeit
- plural: die Gruppenarbeiten

Examples:

Die Hausarbeit ist fertig. – The term paper is finished.
Die Hausarbeiten wurden korrigiert. – The term papers were corrected.
Die Gruppenarbeit macht Spaß. – The group work is fun.
Die Gruppenarbeiten werden morgen präsentiert. – The group projects will be presented tomorrow.

In your sentence:

der Hausarbeit → feminine singular, genitive
die Gruppenarbeit → feminine singular, accusative

Could I say Eine sorgfältige Korrektur der Hausarbeit zeigt, wie du die Gruppenarbeit ernst nimmst instead? Is that correct, and does it change the meaning?

Yes, that sentence is grammatically correct, but the emphasis shifts slightly.

Original:

... zeigt, wie ernst du die Gruppenarbeit nimmst.
- Focus: how serious (degree of seriousness).

Alternative:

... zeigt, wie du die Gruppenarbeit ernst nimmst.
- Grammatically okay, but stylistically less idiomatic.
- wie du ... ernst nimmst sounds more like “in what way / in what manner you take the group work seriously”, which is a bit odd.

Native speakers clearly prefer:

... wie ernst du die Gruppenarbeit nimmst.
or
... dass du die Gruppenarbeit ernst nimmst.

So: your alternative is not wrong, but it sounds less natural and weaker in style.

Can I drop the article and say wie ernst du Gruppenarbeit nimmst instead of wie ernst du die Gruppenarbeit nimmst?

You can, but the meaning changes:

die Gruppenarbeit
- refers to a specific instance of group work
- here: probably the group project you are currently doing
Gruppenarbeit (without article)
- refers more to group work in general, as an abstract concept or as a habitual thing

So:

... wie ernst du die Gruppenarbeit nimmst.
→ how seriously you take this particular group assignment / project.
... wie ernst du Gruppenarbeit nimmst.
→ how seriously you take group work in general.

In the context of a specific course, die Gruppenarbeit is usually more natural.

Why is the main verb zeigt in the present tense? Could this also refer to the future, like “will show”?

German often uses the present tense where English might use present or future, depending on context.

Eine sorgfältige Korrektur der Hausarbeit zeigt, ...

This can mean:

present: A careful correction (in general / usually) shows ...
near future or planned context: A careful correction (that you are going to do) will show ...

German Präsens can cover:

present time:
- Er kommt morgen. – He is coming tomorrow / he will come tomorrow.
future meaning when clearly indicated by context or time expressions.

So your sentence in context might be translated either as:

A careful correction of the term paper shows how seriously you take the group work.
A careful correction of the term paper will show how seriously you take the group work.

but the German form zeigt stays the same.

What is the difference between sorgfältig and sorgfältige here, and how does adjective declension work in this case?

The base form is the adjective sorgfältig = careful / thorough.

When used before a noun, the adjective usually needs an ending that agrees with:

gender (masc/fem/neut)
number (sing/plural)
case (nom/acc/dat/gen)
and the type of article (definite, indefinite, none)

In your sentence:

noun: Korrektur (fem., singular)
case: nominative (subject)
article: eine (indefinite, feminine nominative)

The correct ending for adjectives in this pattern is -e:

eine sorgfältige Korrektur

Other examples:

die sorgfältige Korrektur – definite article, fem. nom. sg.
mit einer sorgfältigen Korrektur – dative, fem. sg. → -en
keine sorgfältige Korrektur – negative article, fem. nom. sg. → -e

So:

sorgfältig = dictionary form / base form
sorgfältige = the form required here by feminine nominative singular with an indefinite article.