To rozwiązanie jest możliwe.

Here to means this.

So To rozwiązanie = this solution.

A useful thing to know is that to is very common in Polish and can mean slightly different things depending on context, such as this, it, or that. In this sentence, it points to a specific thing: this solution.

Because rozwiązanie is a neuter noun, to is the natural form here.

Because rozwiązanie is in the nominative singular, which is the normal case for the subject of the sentence.

In To rozwiązanie jest możliwe, the subject is to rozwiązanie, so nominative is used.

Quick comparison:

• rozwiązanie = nominative singular
• rozwiązania = could be genitive singular or nominative/accusative plural, depending on context
• rozwiązaniem = instrumental singular

Here, the basic dictionary form rozwiązanie is exactly what you need.

Because it has to agree with rozwiązanie, which is neuter singular.

The adjective możliwy changes its ending to match the noun:

• masculine singular: możliwy
• feminine singular: możliwa
• neuter singular: możliwe
• plural (non-masculine personal): możliwe
• plural (masculine personal): możliwi

Since rozwiązanie is neuter singular, the correct form is możliwe.

In standard Polish, jest is normally used in the present tense in this kind of sentence.

So:

• To rozwiązanie jest możliwe = normal, complete sentence

If you say just To rozwiązanie możliwe, it sounds incomplete or very marked in ordinary speech. You might see omission of jest in headlines, notes, or very compressed style, but not as the normal everyday form.

So for learners, the safe rule is:

• keep jest in sentences like this

It can mean more than one thing, depending on context.

Rozwiązanie can mean:

• solution to a problem
• resolution
• in some contexts, the act of solving

In this sentence, it most naturally means solution.

This is common in Polish: a noun derived from a verb can name both the result and, in some contexts, the action.

Yes, Polish word order is more flexible than English, but the version you have is the most neutral and natural.

Default order here:

• To rozwiązanie jest możliwe.

Other orders are possible, but they change emphasis:

• Możliwe jest to rozwiązanie. = more formal or emphatic
• To rozwiązanie możliwe jest. = unusual, poetic, or strongly marked

So for normal use, the original word order is best.

They are not the same.

• To rozwiązanie jest możliwe means that this solution is possible
• To możliwe rozwiązanie means this is a possible solution or this possible solution

The difference is grammatical and semantic:

1. In To rozwiązanie jest możliwe, możliwe is the predicate adjective after jest.
   You are making a statement about the solution.

2. In To możliwe rozwiązanie, możliwe directly modifies rozwiązanie inside the noun phrase.
   You are describing the kind of solution it is.

So English often uses similar words for both, but Polish structures them differently.

Yes. With an adjective like this, Polish uses the nominative.

So:

• To rozwiązanie jest możliwe.

Both rozwiązanie and możliwe are nominative neuter singular.

This is a helpful contrast:

• predicate adjectives after być usually stay in the nominative
• predicate nouns are often instrumental

Compare:

• To rozwiązanie jest możliwe.
• To rozwiązanie jest problemem.

In the second sentence, problemem is instrumental because it is a noun.

A rough pronunciation guide is:

rozwiązanie ≈ roz-vyon-ZA-nye

A few important points:

• w sounds like English v
• ą before z is usually pronounced somewhat like on/om, not like a strong nasal vowel in French
• zanie sounds roughly like ZA-nye
• stress in Polish is usually on the second-to-last syllable, so here the stress is on za:
  roz-wią-ZA-nie

This word can feel long at first, but it becomes much easier once you hear it a few times.

A rough guide is:

możliwe ≈ mozh-LEE-ve

Notes:

• ż sounds like the s in measure
• stress is on the second-to-last syllable: LI
• final -we is pronounced with v, because Polish w sounds like v

So the stress pattern is:

• moż-LI-we

Only the agreeing words would change.

For example:

• masculine: Ten plan jest możliwy.
• feminine: Ta metoda jest możliwa.
• neuter: To rozwiązanie jest możliwe.

So you can see two kinds of agreement:

1. the demonstrative changes: ten / ta / to
2. the adjective changes: możliwy / możliwa / możliwe

That is a very common pattern in Polish, and this sentence is a good example of it.