Cette tache ne part pas, même avec un chiffon propre.

Because tache is a feminine singular noun in French, so it takes the feminine singular demonstrative cette.

• ce = this/that + masculine singular noun
• cette = this/that + feminine singular noun
• ces = these/those + plural noun

So:

• ce chiffon = this cloth
• cette tache = this stain

A useful warning: tache means stain, while tâche with a circumflex usually means task.

Ne ... pas makes the verb negative.

Here the verb is part (from partir in this expression), so:

• ça part = it comes off / it goes away
• ça ne part pas = it does not come off / it won’t come out

In everyday spoken French, people often drop ne, so you may hear:

• Cette tache part pas.

But in standard written French, ne ... pas is expected.

Yes, partir often means to leave, but in some contexts it can also mean to go away, to come off, or to disappear.

With stains, dirt, marks, etc., French often says that the stain goes away:

• La tache part. = The stain comes out.
• La tache ne part pas. = The stain won’t come out.

So this is a very natural French way to talk about whether a stain can be removed.

Because the subject is cette tache, which is third person singular: it.

Present tense of partir:

• je pars
• tu pars
• il / elle / on part
• nous partons
• vous partez
• ils / elles partent

Since cette tache = it, the correct form is part.

French often expresses this idea differently from English.

English commonly says:

• This stain won’t come out.
• I can’t remove this stain.

French often prefers the stain itself as the subject:

• Cette tache ne part pas.
  Literally: This stain doesn’t go away.

If you want a more active version, you could also say:

• Je n’arrive pas à enlever cette tache. = I can’t remove this stain.

Both are possible, but Cette tache ne part pas is very idiomatic.

Même avec means even with.

So:

• avec un chiffon propre = with a clean cloth
• même avec un chiffon propre = even with a clean cloth

It adds the idea that the usual or expected method is not working.

Similar examples:

• Même avec du savon, ça ne part pas. = Even with soap, it won’t come out.
• Même avec de l’aide, il n’y arrive pas. = Even with help, he can’t do it.

In French, most adjectives come after the noun, and propre here means clean, so it normally follows the noun:

• un chiffon propre = a clean cloth

Some adjectives often come before the noun, but propre in the sense of clean usually comes after.

Be careful: propre can also mean own, and then the structure changes:

• son propre chiffon = his/her own cloth

So in your sentence, propre clearly means clean.

Because propre is an adjective describing chiffon, and avec is a preposition meaning with.

The structure is:

• avec = with
• un chiffon = a cloth
• propre = clean

So:

• avec un chiffon propre = with a clean cloth

French does not need anything extra there. It works just like:

• avec un savon doux = with a mild soap
• avec de l’eau chaude = with hot water

No. This is an important spelling distinction:

• tache = stain
• tâche = task, job

So in this sentence, Cette tache ne part pas, it must be tache without the circumflex because we are talking about a stain.

Examples:

• J’ai une tache sur ma chemise. = I have a stain on my shirt.
• J’ai une tâche à finir. = I have a task to finish.

Yes, you might hear that, and it can mean something like This stain won’t go away. But Cette tache ne part pas is especially common and natural when talking about stains, marks, dirt, and similar things that should come off a surface.

So:

• Cette tache ne part pas. = very natural for stains
• Cette tache ne s’en va pas. = understandable, but less standard in this context

If you are learning the most idiomatic everyday phrasing, ne part pas is the better choice here.

In everyday speech, many speakers drop ne, so it may sound like:

• Cette tache part pas, même avec un chiffon propre.

Also, pronunciation matters:

• cette sounds roughly like set
• tache sounds roughly like tash
• part pas sounds like par pa
• même avec sounds roughly like mem avèk

But in writing, especially careful writing, keep:

• Cette tache ne part pas, même avec un chiffon propre.