Он поинтересовался, есть ли у меня ещё один аргумент против этой поездки.

Both can mean asked, but поинтересовался is a bit softer and more nuanced. It literally suggests showed interest / inquired rather than simply asked.

So:

• Он спросил... = He asked...
• Он поинтересовался... = He inquired / He asked with interest...

In many contexts, поинтересовался can sound slightly more polite, less blunt, or more formal than спросил.

Ли is a particle used to introduce an indirect yes/no question.

In this sentence:

• есть ли у меня... = whether I have... / if I have...

So the structure is:

• Он поинтересовался, есть ли...
• He asked whether / if there is...

This is one of the most common uses of ли in Russian.

Ли usually comes right after the word it relates to most directly. Here it relates to есть, so:

• есть ли = whether there is / whether I have

That is the normal order.

Compare:

• Я не знаю, придёт ли он. = I don’t know whether he will come.
• Он спросил, есть ли время. = He asked whether there was time.

So ли is often placed after the verb in indirect yes/no questions.

Russian usually expresses possession with the structure:

• у + genitive + есть

So:

• у меня есть literally means something like at me there is
• natural English translation: I have

In your sentence:

• есть ли у меня ещё один аргумент = whether I have another argument

The form меня is genitive because it follows у.

Because after the preposition у, Russian requires the genitive case.

So:

• я = nominative, I
• меня = genitive/accusative form of я

In the possession pattern:

• у меня есть... = I have...
• у него есть... = he has...
• у нас есть... = we have...

You cannot say есть ли я ещё один аргумент here, because that is not how possession is expressed in standard Russian.

Here ещё один means one more / another.

So:

• ещё один аргумент = another argument or one more argument

Be careful: ещё can mean different things depending on context:

• ещё = still
  
  • Он ещё дома. = He is still at home.
• ещё = more / another
  
  • ещё один = one more / another

In this sentence, the second meaning is the correct one.

Both can sometimes be translated as another, but they are not identical.

• ещё один = one more, an additional one
• другой = a different one, another one of a different kind

So:

• ещё один аргумент = one more argument
• другой аргумент = a different argument

In your sentence, the idea is probably Do I have one more argument against the trip?, so ещё один fits better.

The preposition против requires the genitive case.

So:

• эта поездка = nominative, this trip
• этой поездки = genitive, used after против

That is why we get:

• аргумент против этой поездки = an argument against this trip

This is a very important pattern to remember:

• против кого? чего?
  • genitive

Examples:

• против плана = against the plan
• против идеи = against the idea
• против этой поездки = against this trip

Yes, but in Russian аргумент often means reason, point, or supporting consideration, not just a verbal quarrel.

So in this sentence:

• аргумент против этой поездки is more naturally understood as
  • an argument against this trip
  • a reason not to go on this trip
  • a point against this trip

It does not mean that someone is fighting or having an angry dispute.

Because the sentence contains a main clause and a subordinate clause.

• Он поинтересовался = main clause
• есть ли у меня ещё один аргумент против этой поездки = subordinate clause

Russian normally separates such clauses with a comma.

So the structure is:

• He inquired, whether/if I had another argument against the trip.

In Russian punctuation, that comma is required.

To some extent, yes. Russian word order is flexible, but not completely free.

The neutral order here is:

• есть ли у меня ещё один аргумент против этой поездки

You might also see slight variations for emphasis, for example:

• есть ли у меня аргумент против этой поездки
• есть ли ещё один аргумент у меня — possible, but less natural here

In this sentence, the given word order sounds natural and standard.

Поинтересовался is:

• past tense
• masculine singular
• perfective

The infinitive is поинтересоваться.

Why perfective? Because it refers to a completed single act of asking/inquiring:

• Он поинтересовался... = He asked / He inquired...

If you used the imperfective интересовался, it would usually mean something more like:

• He was asking about
• He was interested in
• He kept asking about

So поинтересовался is the natural choice for a one-time completed question.

A direct version could be:

• У тебя есть ещё один аргумент против этой поездки?
• У вас есть ещё один аргумент против этой поездки?

That means:

• Do you have another argument against this trip?

When it becomes indirect after a verb like поинтересовался, Russian often uses ли:

• Он поинтересовался, есть ли у меня ещё один аргумент против этой поездки.

So you can think of it as the direct yes/no question being turned into an indirect whether/if clause.

In the present tense, Russian sometimes omits есть, but in this kind of sentence есть is very natural and often necessary for clarity.

Compare:

• У меня есть аргумент. = I have an argument.
• У меня аргумент. = possible in some contexts, but much less neutral and not the best choice here

With ли, the verb есть is especially important:

• есть ли у меня... = whether I have...

So in this sentence, keeping есть is the normal and correct choice.