Multum interest utrum cives bonum commune an commodum privatum quaerant.

It means it matters a great deal or it makes a big difference.

A few useful points:

• interest is often used impersonally in Latin.
• So Latin is not saying that some person or thing is the subject of interest.
• Instead, the whole idea is: it matters.

Here multum means much / greatly, so multum interest = it matters greatly.

Interest is singular because it is being used as an impersonal verb.

In English, this is similar to expressions like:

• it matters
• it is important
• it makes a difference

The real content of what matters is expressed by the clause after it:

utrum cives bonum commune an commodum privatum quaerant

So the sense is:

• It matters a great deal [whether the citizens seek the common good or private advantage].

That whole whether-clause is the thing that matters.

Utrum ... an ... means whether ... or ...

It introduces an indirect question with two alternatives:

• utrum = whether
• an = or

So here the two alternatives are:

• bonum commune quaerant = they seek the common good
• commodum privatum quaerant = they seek private advantage

Latin often uses this pair when presenting a choice between two possibilities.

Because in this construction an is not the common word for and.
Here it is part of the pair utrum ... an ..., which means whether ... or ...

So:

• utrum A an B = whether A or B

That is a very common Latin pattern.

Because it is in an indirect question introduced by utrum.

In Latin, verbs in indirect questions normally go into the subjunctive.

So:

• direct question: Utrum cives bonum commune an commodum privatum quaerunt?
• indirect question: Multum interest utrum cives bonum commune an commodum privatum quaerant.

The subjunctive quaerant is therefore exactly what we expect.

It is present subjunctive because the main verb is present (interest) and the action in the indirect question is viewed as contemporaneous with it.

Very roughly:

• interest = it matters
• quaerant = whether they are seeking / seek

This follows the normal sequence of tenses in Latin:

• primary tense in the main verb (interest is present)
• present or perfect subjunctive in the subordinate clause, depending on meaning

Here the present subjunctive fits the sense best.

Cives is nominative plural.

It is the subject of quaerant:

• cives quaerant = the citizens seek

Even though the whole sentence begins with impersonal interest, inside the utrum clause we still need a normal subject for the verb quaerant, and that subject is cives.

They are both accusative singular and they are the direct objects of quaerant.

So:

• bonum commune = the common good
• commodum privatum = private advantage

And:

• cives bonum commune quaerant = the citizens seek the common good
• cives commodum privatum quaerant = the citizens seek private advantage

The adjectives agree with their nouns:

• bonum with bonum
• commune with bonum
• commodum with commodum
• privatum with commodum

All are neuter singular accusative.

Because the adjectives must agree with the nouns they describe in gender, number, and case.

Here both nouns are neuter singular accusative:

• bonum
• commodum

So their adjectives must also be neuter singular accusative:

• commune (from communis, commune)
• privatum (from privatus, -a, -um)

A learner often notices that commune does not look like bonum. That is because communis is a third-declension adjective, so its neuter nominative/accusative singular is commune, not communum or something similar.

Latin often omits a repeated word when it is easily understood.

So the sentence literally gives:

• whether the citizens seek the common good
• or private advantage

The second quaerant is simply understood:

• utrum cives bonum commune quaerant an commodum privatum [quaerant]

English does the same thing:

• whether citizens seek the common good or private advantage

We do not need to repeat seek.

Yes, quaerere can mean several related things, including:

• to ask
• to inquire
• to look for
• to seek
• to aim at / pursue

In this sentence it clearly means seek or pursue in a moral or political sense.

So bonum commune quaerere means something like:

• to seek the common good
• to aim at the common good
• to pursue the common good

Formally, multum is the neuter singular accusative of multus, but here it is being used adverbially.

So it means:

• much
• greatly
• a lot

This is very common in Latin. A neuter accusative form can function like an adverb.

So:

• multum interest = it matters a lot

Yes, Latin word order is relatively flexible.

This order is effective because:

• Multum interest comes first and immediately states the main point: it matters a great deal
• utrum then introduces the issue that matters
• bonum commune and commodum privatum are placed in a neat contrast

So the sentence is carefully balanced:

• bonum commune
• commodum privatum

A different order could still be grammatical, but this version is clear and rhetorically strong.