Mi bezonas plian dentokontrolon.

• Mi = I
• bezonas = need
• plian = additional / another / one more
• dentokontrolon = dental check-up or literally tooth-check

So the sentence means something like I need another dental check-up or I need an additional dental examination.

In Esperanto, -as marks the present tense.

So:

• bezoni = to need
• bezonas = need / am needing

The subject is Mi, so Mi bezonas means I need.

Unlike English, Esperanto verbs do not change according to the person:

• mi bezonas = I need
• vi bezonas = you need
• li bezonas = he needs

The verb form stays the same.

The -n ending marks the direct object in Esperanto.

In this sentence, the thing being needed is plian dentokontrolon, so that whole noun phrase is the direct object.

• Mi bezonas kion? = What do I need?
• Answer: plian dentokontrolon

Because dentokontrolon is the noun being acted on, it gets -n.
And because plian describes that noun, the adjective must match it, so it also gets -n.

This is called agreement:

• plia dentokontrolo = an additional dental check-up
• plian dentokontrolon = an additional dental check-up (as a direct object)

pli by itself is an adverb meaning more.

But here we need an adjective describing the noun dentokontrolon, so Esperanto uses plia, meaning additional or one more.

Compare:

• Mi bezonas pli da tempo. = I need more time.
  Here pli modifies the idea of quantity.

• Mi bezonas plian kontrolon. = I need another/additional check-up.
  Here plian modifies the noun kontrolon.

So plian is the correct form because it is acting like an adjective.

This is a very common question.

• plia = additional, one more
• alia = other, different, another one of a different kind

So:

• Mi bezonas plian dentokontrolon.
  = I need one more dental check-up.

• Mi bezonas alian dentokontrolon.
  = I need a different dental check-up / another kind of dental check-up.

In many English sentences, another can mean either one, but Esperanto often separates the ideas more clearly. Here plian is the natural choice if the meaning is an additional check-up.

It is a compound word:

• dent- from dento = tooth
• kontrolo = check, inspection, examination

Together:

• dentokontrolo = dental check-up / tooth examination

Then it gets -n because it is the direct object:

• dentokontrolon

This is very normal in Esperanto. The final part of the compound carries the main grammatical ending:

• dompordo = house door
• manĝotablo = dining table
• dentokontrolo = dental check-up

Esperanto has no indefinite article.

English says:

• a dental check-up
• an additional check-up

Esperanto usually just says:

• dentokontrolo
• plia dentokontrolo

Esperanto does have a definite article, la, meaning the. But if English would use a/an, Esperanto normally uses nothing.

So:

• Mi bezonas dentokontrolon. = I need a dental check-up.
• Mi bezonas la dentokontrolon. = I need the dental check-up.

Yes, it could come after.

Both are grammatically possible:

• plian dentokontrolon
• dentokontrolon plian

But putting the adjective before the noun is much more common and usually sounds more natural, especially in simple everyday sentences.

Also, the adjective still has to agree with the noun in ending:

• plia dentokontrolo
• plian dentokontrolon

Whether it comes before or after, the matching endings stay the same.

There can be other ways to express the idea, depending on context.

For example, speakers might also say something like:

• denta kontrolo = dental check-up
• kontrolo ĉe la dentisto = a check-up at the dentist

But dentokontrolo is a perfectly understandable compound and follows normal Esperanto word-building.

A learner should notice an important pattern here: Esperanto often prefers building clear compounds instead of relying on separate fixed phrases. That makes words like dentokontrolo very natural.