Avec l’aide de Paul, je peux finir le travail plus vite.

They do not mean exactly the same thing.

• Avec l’aide de Paul = with Paul’s help
• Avec Paul = with Paul / together with Paul

So Avec l’aide de Paul emphasizes that Paul is helping you.
Avec Paul could simply mean Paul is present or working alongside you.

It literally means the help of Paul, which in natural English becomes Paul’s help.

This is a very common French pattern:

• le livre de Marie = Marie’s book
• l’idée de Jean = Jean’s idea
• l’aide de Paul = Paul’s help

French often uses de where English uses ’s.

Because aide begins with a vowel sound, la becomes l’ before it.

So:

• la aide → not used
• l’aide → correct

This is called elision. It happens with articles like le and la before a vowel or silent h:

• l’homme
• l’école
• l’aide

Here, de links aide to the person giving the help.

• l’aide de Paul = Paul’s help
• l’aide de ma sœur = my sister’s help

This is one of the most common uses of de in French: showing possession or connection.

Because after pouvoir (can / to be able to), French uses an infinitive.

So:

• je peux finir = correct
• je peux finis = incorrect

This is the same idea as in English:

• I can finish
• not I can finished

Other examples:

• je peux venir = I can come
• je peux travailler = I can work

Peux is the je form of pouvoir in the present tense.

Present tense of pouvoir:

• je peux
• tu peux
• il/elle peut
• nous pouvons
• vous pouvez
• ils/elles peuvent

So je peux finir means I can finish.

Usually, no—not in this sentence.

For je, French has:

• je peux = normal everyday form
• puis-je = formal inversion, mainly in questions

So you say:

• Je peux finir le travail. = correct
• Puis-je finir le travail ? = May/Can I finish the work?

You would not normally say je puis finir in modern everyday French.

French usually needs an article where English may not.

So French says:

• finir le travail

even where English might simply say:

• finish the work
• or sometimes just finish work, depending on context

Here le travail refers to the work / the task / the job being talked about.

Plus vite means faster / more quickly.

• vite = quickly / fast
• plus vite = more quickly / faster

It comes after the verb phrase because French often places adverbial expressions like this after the verb or after the object:

• Je travaille vite.
• Je finis le travail plus vite.

That word order is very natural in French.

Yes. Both are correct.

• plus vite = faster
• plus rapidement = more quickly

Plus vite is usually more common and more conversational.
Plus rapidement can sound a bit more formal or more explicit.

So:

• je peux finir le travail plus vite
• je peux finir le travail plus rapidement

Both work.

Yes, in many situations.

• finir le travail
• terminer le travail

Both can mean finish the work.

A small nuance:

• finir is a very common general verb for finishing
• terminer can sound a little more like bring to completion

In this sentence, either one would sound natural.

It is a very common expression, yes. It means with the help of ...

You can reuse the pattern with other nouns:

• Avec l’aide de ma sœur...
• Avec l’aide du professeur...
• Avec l’aide de mes amis...

So it is a useful structure to learn as a chunk: avec l’aide de + person/thing

The comma separates the introductory phrase from the main clause.

• Avec l’aide de Paul, = introductory phrase
• je peux finir le travail plus vite. = main clause

In French, this comma is natural and helps readability, especially when the sentence begins with a prepositional phrase. It is not always absolutely required in short sentences, but it is very common here.