Ohne Kofferanhänger finde ich mein Aufgabegepäck nicht.

Because German uses the verb-second rule in main clauses.

Here, Ohne Kofferanhänger is placed first for emphasis. Once that first slot is taken, the conjugated verb must still come second:

• Ohne Kofferanhänger | finde | ich mein Aufgabegepäck nicht.

If you started with the subject instead, you would get:

• Ich finde mein Aufgabegepäck ohne Kofferanhänger nicht.

That is also possible, but it shifts the emphasis a bit.

Ohne always takes the accusative case.

So after ohne, you would expect an accusative noun phrase:

• ohne einen Kofferanhänger
• ohne den Kofferanhänger

In your sentence, the article is omitted, so you do not see the case marking directly:

• Ohne Kofferanhänger

But the case is still accusative because ohne requires it.

German often omits the article when speaking in a general or indefinite way, especially in expressions like this.

So Ohne Kofferanhänger means something like:

• without a baggage tag
• without any baggage tag

If you add an article, the meaning becomes more specific:

• ohne einen Kofferanhänger = without a baggage tag / without one baggage tag
• ohne den Kofferanhänger = without the baggage tag (a specific one already known)

The version without an article sounds broad and natural here.

Because Aufgabegepäck is neuter: das Aufgabegepäck.

After finden, the object is in the accusative case. For a neuter singular noun, the possessive determiner looks like this:

• nominative: mein Aufgabegepäck
• accusative: mein Aufgabegepäck

So there is no extra ending here.

Compare:

• mein Koffer → masculine nominative
• meinen Koffer → masculine accusative

But with a neuter noun like Gepäck, it stays mein.

In this sentence, nicht negates the whole verbal idea: I do not find ...

German often puts nicht toward the end of the clause when it negates the main action or a large part of the sentence.

So:

• Ohne Kofferanhänger finde ich mein Aufgabegepäck nicht.

This is a very normal placement.

If nicht were placed earlier, it could change the focus of the negation. German uses position to show what exactly is being negated.

No. In this word, Aufgabe does not mean homework or task.

Here it comes from the verb aufgeben, which can mean to check in or to hand over luggage at the airport.

So Aufgabegepäck means:

• baggage that has been checked in
• checked baggage

This is a good example of how a familiar German word can have a different meaning inside a compound.

German often combines nouns into one long word.

Two useful rules:

1. The last part is the main noun.
2. The last part decides the gender.

So:

• Kofferanhänger = Koffer

  • Anhänger
    Literally, a suitcase tag / luggage tag
    Since der Anhänger is masculine, der Kofferanhänger is masculine.

• Aufgabegepäck = Aufgabe

  • Gepäck
    Since das Gepäck is neuter, das Aufgabegepäck is neuter.

This is one of the most important patterns in German vocabulary.

It is understandable and grammatical, but some speakers might choose slightly different vocabulary depending on the context.

For example, at an airport you may also hear:

• Gepäckanhänger for baggage tag
• aufgegebenes Gepäck for checked baggage

So a more idiomatic version might be:

• Ohne den Gepäckanhänger finde ich mein aufgegebenes Gepäck nicht.

That said, your original sentence is still clear and works well for learning the grammar.

Because all nouns are capitalized in German.

That includes:

• Kofferanhänger
• Aufgabegepäck

This is one of the easiest spelling rules in German to notice, even though it takes time to apply consistently when writing.