Hanno respinto la prima soluzione.

It is the passato prossimo, a very common Italian past tense used for completed actions.

It is formed like this:

• hanno = present tense of avere for they
• respinto = past participle of respingere

So hanno respinto literally means they have rejected, but in natural English it can often be translated as either:

• They rejected
• They have rejected

The best choice depends on context.

Hanno is the 3rd person plural form of avere in the present tense, meaning they have.

In this sentence, it is not being used with its full meaning of possession. Instead, it works as an auxiliary verb to build the passato prossimo:

• hanno respinto = they have rejected / they rejected

Italian often leaves out the subject pronoun, so loro is not stated:

• (Loro) hanno respinto la prima soluzione.

Because respingere normally takes avere as its auxiliary in compound tenses.

A quick rule:

• Most transitive verbs (verbs that take a direct object) use avere
• Many intransitive verbs of movement/change of state use essere

Here the verb has a direct object:

• la prima soluzione

So Italian uses:

• hanno respinto

not:

• sono respinti

Also, sono respinti would usually mean something like they are rejected or they were rejected, which is a different structure.

Yes. The infinitive is respingere and its past participle is respinto, which is irregular.

So:

• respingere = to reject, repel
• respinto = rejected

You cannot form it regularly as something like respingito.

This pattern is related to other verbs with irregular participles, for example:

• scrivere → scritto
• rompere → rotto
• spingere → spinto

Since respingere contains spingere, the participle respinto is easier to remember.

Because with avere, the past participle usually does not agree with the direct object that comes after it.

So Italian says:

• Hanno respinto la prima soluzione.

not:

• Hanno respinta la prima soluzione.

The participle stays in the default form respinto.

However, agreement can happen when the direct object comes before the verb as a pronoun:

• La prima soluzione? L'hanno respinta.

Here respinta agrees with la / la prima soluzione, which is feminine singular.

It means the first solution.

Breakdown:

• la = the (feminine singular definite article)
• prima = first (feminine singular)
• soluzione = solution (feminine singular noun)

Everything matches in gender and number:

• la → feminine singular
• prima → feminine singular
• soluzione → feminine singular

Because prima here is an ordinal number meaning first, and ordinals normally go before the noun in Italian:

• la prima soluzione
• il secondo capitolo
• la terza volta

So this word order is completely standard.

In this sentence, la is needed because the phrase refers to a specific item: the first solution.

Italian uses definite articles very naturally in places where English also uses them when the thing is specific:

• la prima soluzione = the first solution

Without the article, prima soluzione would sound less complete in this sentence and would usually need a different structure or context.

The subject is understood from the verb form hanno, so Italian does not need to say loro.

Italian is a pro-drop language, which means subject pronouns are often omitted when the verb ending already makes the subject clear.

So:

• Hanno respinto la prima soluzione. and
• Loro hanno respinto la prima soluzione.

both mean the same thing, but the version without loro is more natural unless you want emphasis or contrast.

Yes. That is very common with the passato prossimo.

Italian hanno respinto can correspond to:

• They rejected the first solution
• They have rejected the first solution

English chooses between those two based on style and context, but Italian often uses the same form for both.

Yes, Hanno respinto la prima soluzione is a normal, neutral word order.

It is basically:

• [verb] + [object] with the subject omitted

You can change the order for emphasis, but then the tone changes. For example:

• La prima soluzione l'hanno respinta. = It was the first solution that they rejected.

That version gives extra emphasis to la prima soluzione. The original sentence is the most straightforward and neutral one.