Brinem o kućnom ljubimcu.

Here brinem means I take care of or I care for.

It is the 1st person singular present tense of brinuti. So:

• brinem = I care for / I am taking care of
• the subject I is already built into the verb form

Depending on context, present tense in Serbian can sound like:

• I am taking care of the pet
• I take care of the pet

Because Serbian often drops subject pronouns when they are clear from the verb.

• brinem already tells you the subject is I
• so Ja brinem o kućnom ljubimcu is possible, but ja is not necessary

You would usually add ja only for emphasis, contrast, or clarity:

• Ja brinem o kućnom ljubimcu, a on ne.
  I take care of the pet, but he doesn’t.

The preposition o is part of this expression.

With brinuti o + locative, the meaning is to care for / to look after someone or something.

So:

• brinuti o detetu = to take care of a child
• brinuti o kućnom ljubimcu = to take care of a pet

This is different from some other meanings of brinuti, where the construction may change.

Kućnom ljubimcu is in the locative singular.

The basic dictionary form is:

• kućni ljubimac = pet

After o, Serbian uses the locative, so the words change form:

• kućni → kućnom
• ljubimac → ljubimcu

So:

• kućni ljubimac = nominative
• o kućnom ljubimcu = locative after o

This kind of change is completely normal in Serbian and is one of the main things English speakers have to get used to.

In this sentence, it most naturally means I take care of the pet or I care for the pet.

That is because brinuti o + locative usually expresses care/responsibility.

If you want to say I am worried about the pet, Serbian often uses a different pattern, for example:

• Brinem za kućnog ljubimca.

So a useful contrast is:

• brinem o ... = I take care of ...
• brinem za ... = I worry about ...

Context can still matter, but this distinction is very helpful.

Because Serbian does not have articles like English a/an and the.

So kućnom ljubimcu could mean:

• a pet
• the pet
• sometimes even my pet, if the context makes that clear

If you want to be more specific, Serbian usually adds another word:

• o jednom kućnom ljubimcu = about one / a certain pet
• o tom kućnom ljubimcu = about that pet
• o svom kućnom ljubimcu = about my own pet / my pet

Yes. Serbian word order is more flexible than English word order.

The neutral order here is:

• Brinem o kućnom ljubimcu.

But other orders are possible for emphasis, for example:

• O kućnom ljubimcu brinem.

That version puts more focus on the pet.

Even though word order can move around, the case endings still show how the words relate to each other, so the meaning stays clear.

The dictionary form is kućni ljubimac.

That is the form you would look up in a dictionary.

It is made of:

• kućni = domestic / household
• ljubimac = pet / favorite animal

Together, kućni ljubimac means pet.

In everyday speech, people may also simply say ljubimac when the meaning is obvious.

It is a normal Serbian expression for pet.

Literally, it is close to household pet or domestic pet, but in actual use it simply means pet.

So English speakers should not over-translate it word by word. In most cases:

• kućni ljubimac = pet

That is the natural meaning to remember.