A hosszabbítás és az előjegyzés gyorsak.

In Hungarian, the present-tense verb to be is usually left out in the 3rd person when the predicate is a noun or an adjective.

So:

• A hosszabbítás és az előjegyzés gyorsak.
• literally: The extension and the reservation quick
• natural English: The extension and the reservation are quick

Hungarian does not need vannak here.

Compare:

• A hosszabbítás gyors. = The extension is quick.
• A hosszabbítás gyors volt. = The extension was quick.

In the past tense, the verb appears again: volt.

They are not different in meaning. Both mean the.

The choice depends on the sound that follows:

• a before a consonant sound
• az before a vowel sound

So here:

• a hosszabbítás because h is a consonant
• az előjegyzés because e is a vowel

This is purely a pronunciation-related rule, similar to how English uses a and an.

Because Hungarian normally repeats the definite article before each coordinated noun.

So the natural pattern is:

• a hosszabbítás és az előjegyzés

This is like saying:

• the extension and the reservation

For learners, it is best to think of each noun as getting its own article in this kind of structure.

Each noun is singular by itself:

• hosszabbítás = one extension / renewal
• előjegyzés = one reservation / booking / appointment

But together, joined by és meaning and, they form a compound subject, which is treated as plural overall.

So:

• A hosszabbítás gyors. = The extension is quick.
• Az előjegyzés gyors. = The reservation is quick.
• A hosszabbítás és az előjegyzés gyorsak. = The extension and the reservation are quick.

Also, both nouns are in the basic nominative form here, because they are the subject of the sentence.

Because gyorsak is the plural form of the adjective gyors when it is used as the predicate.

• gyors = quick, fast
• gyorsak = quick, fast, with a plural subject

Since the subject is A hosszabbítás és az előjegyzés = the extension and the reservation, the predicate adjective becomes plural:

• gyorsak

A useful thing to remember:

• Attributive adjective before a noun: usually no plural marking on the adjective
  
  • gyors hosszabbítások = quick renewals
• Predicative adjective after the subject: plural marking appears
  
  • A hosszabbítások gyorsak. = The renewals are quick.

The ending -ak is used because of vowel harmony. With other adjectives, you may see -ek or -ok instead.

Because here gyorsak is not directly describing a noun inside a noun phrase. It is the predicate of the sentence.

That means the structure is:

• subject: A hosszabbítás és az előjegyzés
• predicate: gyorsak

So the sentence means:

• The extension and the reservation are quick

If gyors were placed directly before a noun, it would mean something like:

• a gyors hosszabbítás = the quick extension

That is a different structure. In your sentence, gyorsak is not attached to just one noun; it says something about the whole subject.

Yes. Hungarian word order is more flexible than English word order, and it often reflects focus or emphasis rather than strict grammatical roles.

The version you have:

• A hosszabbítás és az előjegyzés gyorsak.

is a neutral, straightforward statement.

If you move things around, the meaning stays similar, but the emphasis can change. For example, putting gyorsak earlier can highlight the idea of being quick.

For learners, the given order is the safest and most neutral one.

Both are typical Hungarian derived nouns, built from verbs.

• hosszabbítás
  comes from hosszabbít = to extend, to prolong
  plus a noun-forming ending, giving the meaning extension / renewal / prolongation

• előjegyzés
  comes from előjegyez = to reserve, to register in advance, to book
  plus -és, giving the meaning reservation / booking / appointment

This is very common in Hungarian: a verb + a noun-forming suffix creates the name of an action or process.

So these long words are not random; they are built quite systematically.

és means and.

It links the two nouns:

• a hosszabbítás = the extension / renewal
• az előjegyzés = the reservation / booking
• a hosszabbítás és az előjegyzés = the extension and the reservation

Because és joins two separate things, the whole subject becomes plural in meaning, which is why the predicate is gyorsak rather than gyors.