Jeg troede, at opgaven var kompliceret, men lærerens forklaring gjorde den mindre svær.

Troede is the past tense of tro (to think / to believe).

• Jeg tror = I think
• Jeg troede = I thought

The sentence is talking about what the speaker thought earlier, so Danish uses the past tense.

Here at means that and introduces a subordinate clause:

• Jeg troede, at opgaven var kompliceret
  = I thought that the assignment was complicated

In English, that is often optional. In Danish, at is very commonly used in this kind of sentence.

Opgave means assignment, task, or exercise, depending on context.

The ending -en makes it definite:

• en opgave = an assignment / a task
• opgaven = the assignment / the task

So opgaven means the assignment.

Var is the past tense of være (to be), so:

• er kompliceret = is complicated
• var kompliceret = was complicated

Because the speaker is describing a past opinion, Danish uses var.

Also, kompliceret stays the same here because it is describing a singular noun, opgaven.

Yes, there is a small difference.

• kompliceret = complicated, meaning something has many parts or is complex
• svær = difficult / hard, meaning something is not easy to understand or do

So the sentence moves from one idea to a related one:

• the assignment seemed complicated
• the teacher’s explanation made it less difficult

They are similar, but not exactly identical.

Lærerens forklaring is the normal Danish way to say the teacher’s explanation.

The -s on læreren shows possession:

• læreren = the teacher
• lærerens = the teacher’s

So:

• lærerens forklaring = the teacher’s explanation

This possessive structure is usually more natural in Danish than using something like the explanation of the teacher.

To show possession in Danish, you usually add -s to the noun:

• læreren = the teacher
• lærerens = the teacher’s

This works much like English ’s, except Danish just adds s without an apostrophe in normal spelling.

Gjorde is the past tense of gøre (to do / to make).

In this sentence, gøre means to make / to cause something to become:

• gjorde den mindre svær
  literally: made it less difficult

So the structure is:

• X gjorde Y mindre svær
  = X made Y less difficult

Here:

• lærerens forklaring = the thing causing the change
• den = the assignment
• mindre svær = less difficult

Den refers back to opgaven.

Because opgave is a common-gender noun (en opgave), the pronoun used is den:

• opgaven = the assignment
• den = it

So gjorde den mindre svær means made it less difficult, where it = the assignment.

Mindre svær means less difficult, and it is completely natural Danish.

• svær = difficult
• mindre svær = less difficult

You could also express a similar idea with lettere (easier), for example:

• gjorde den lettere = made it easier

The sentence chooses mindre svær because it matches the idea of reducing difficulty rather than directly saying easier. Both are possible, but mindre svær is very close to the English wording less difficult.

Because mere means more, while mindre means less.

• mere svær would mean more difficult (though sværere is usually more natural for more difficult)
• mindre svær means less difficult

Since the explanation improved the situation, Danish uses mindre, not mere.

The second clause is:

• men lærerens forklaring gjorde den mindre svær

This follows normal Danish main-clause word order:

1. men = but
2. lærerens forklaring = subject
3. gjorde = verb
4. den = object
5. mindre svær = complement

So literally:

• but the teacher’s explanation made it less difficult

This is very similar to English word order.

Sometimes in spoken or informal Danish, at can be omitted after verbs like tro, especially if the sentence is simple:

• Jeg troede, opgaven var kompliceret

But in careful, standard written Danish, at is very common and usually preferred:

• Jeg troede, at opgaven var kompliceret

So yes, omission is possible in some contexts, but keeping at is the safest choice for learners.

Yes, it is grammatical and natural.

It sounds like clear, standard written Danish. A native speaker might also choose slightly different wording, for example:

• Jeg troede, at opgaven var kompliceret, men lærerens forklaring gjorde den lettere.

That version is perhaps a bit more idiomatic because lettere is very common, but your original sentence is still perfectly good Danish.