Mater arbitratur vitam publicam difficiliorem esse quam vitam privatam.

The sentence has two parts:

• Mater arbitratur = Mother thinks / judges / considers
• vitam publicam difficiliorem esse quam vitam privatam = the thing she thinks

So the whole sentence is:

• main clause: Mater arbitratur
• indirect statement: vitam publicam difficiliorem esse quam vitam privatam

This is a very common Latin pattern.

Mater is in the nominative singular, which is the case normally used for the subject of a sentence.

So mater is the person doing the thinking:

• mater = mother
• arbitratur = thinks / considers

Also, Latin has no word for the or a, so mater can mean either mother or the mother, depending on context.

Because arbitratur comes from arbitror, which is a deponent verb.

A deponent verb:

• has passive-looking forms
• but an active meaning

So:

• arbitratur looks like a passive form
• but means she thinks, she judges, or she considers

It does not mean she is thought here.

Because after verbs like arbitratur, Latin often uses an indirect statement construction, also called accusative + infinitive.

In this construction:

• the subject of the reported idea goes into the accusative
• the verb of the reported idea goes into the infinitive

So instead of saying:

• Mother thinks that public life...

Latin says:

• Mother thinks public life to be...

That is why vita publica becomes vitam publicam.

In an indirect statement, a normal finite verb like est becomes an infinitive.

So if the direct statement were:

• Vita publica difficilior est quam vita privata.
• Public life is more difficult than private life.

then after arbitratur, Latin changes it to:

• vitam publicam difficiliorem esse quam vitam privatam

So esse is the infinitive corresponding to est.

Because difficiliorem agrees with vitam publicam.

Since vitam publicam is:

• feminine
• singular
• accusative

the adjective describing it must match it. So difficiliorem is also feminine singular accusative in agreement.

In indirect statement, predicate adjectives often match the accusative subject this way.

Because comparative adjectives in Latin belong to the third declension, not the first/second declension pattern you may expect from adjectives like publicam or privatam.

So:

• difficilis = difficult
• difficilior = more difficult
• difficiliorem = more difficult, accusative singular masculine or feminine

That means a feminine noun can perfectly well take difficiliorem. The form is correct because comparatives decline differently.

It is built into difficiliorem.

Latin usually does not add a separate word for more before a comparative adjective. Instead, the adjective itself changes form:

• difficilis = difficult
• difficilior / difficilius = more difficult

So difficiliorem already means more difficult.

Because quam is introducing the second thing being compared, and after quam, Latin normally uses the same case as the first thing being compared.

Here the first thing is:

• vitam publicam — accusative

So the second thing is also accusative:

• quam vitam privatam

If this were a direct statement, you would have:

• vita publica difficilior est quam vita privata

But once the whole statement is turned into indirect statement, the accusative carries through.

Yes. Latin often has two ways to express comparison:

• quam
  • the same case
• the ablative of comparison

So Latin could also say:

• Mater arbitratur vitam publicam difficiliorem esse vitā privatā.

Both are possible. The version with quam vitam privatam is often easier for learners to recognize, because the comparison is stated very explicitly.

You do not need to supply a separate verb in order to understand it. Quam vitam privatam simply means than private life as the second term of comparison.

A helpful way to see it is to compare it with the direct statement:

• Vita publica difficilior est quam vita privata.

Then the indirect statement keeps the same comparison, but shifts the first part into accusative + infinitive:

• vitam publicam difficiliorem esse quam vitam privatam

So learners should read it simply as public life to be more difficult than private life.

Because Latin has no articles.

So:

• mater can mean mother or the mother
• vitam publicam can mean public life
• vitam privatam can mean private life

Context tells you whether English should use a, the, or no article at all.

The word order is natural, but Latin word order is fairly flexible.

This sentence puts the main clause first:

• Mater arbitratur

and then gives the whole indirect statement after it:

• vitam publicam difficiliorem esse quam vitam privatam

That is a clear and common arrangement. But Latin could rearrange some of these words for emphasis without changing the basic meaning.