Hon försöker trösta barnet med en bok och en varm filt.

This sentence follows the normal Swedish main-clause pattern:

• Hon = subject
• försöker = finite verb
• trösta = infinitive verb
• barnet = object
• med en bok och en varm filt = prepositional phrase

So the structure is basically:

Subject + finite verb + infinitive + object + extra information

That is very similar to English She is trying to comfort the child with a book and a warm blanket, except Swedish does not use to before trösta here.

Because försöker means tries / is trying, and trösta means to comfort.

Together they form the idea tries to comfort:

• försöker = present tense of försöka
• trösta = infinitive

This is very common in Swedish:

• Hon försöker sova = She is trying to sleep
• Jag försöker förstå = I am trying to understand

So försöker trösta works just like English tries to comfort.

After försöka, Swedish often leaves out att.

So both of these are possible:

• Hon försöker trösta barnet
• Hon försöker att trösta barnet

The version without att is very natural and common. For a learner, it is safest to recognize that försöka can be followed directly by an infinitive.

Barnet is the definite singular form of barn.

• ett barn = a child
• barnet = the child

Swedish usually expresses definiteness by adding an ending to the noun:

• en bok → boken
• ett barn → barnet

So trösta barnet means comfort the child, not just comfort a child.

Because Swedish has two genders for nouns, and definite forms work differently from indefinite ones.

Here you have:

• bok is a common-gender noun → en bok
• barn is a neuter noun → ett barn, barnet
• filt is also a common-gender noun → en filt

So:

• en is used with common-gender nouns
• ett is used with neuter nouns
• the definite form is usually added as an ending: -en, -et, etc.

That is why the sentence contains both en and -et forms.

Because adjective endings in Swedish change depending on the noun form.

Here, filt is:

• singular
• indefinite
• common gender

In that pattern, the adjective usually has its basic form:

• en varm filt = a warm blanket

Compare:

• ett varmt täcke = a warm quilt/cover
• den varma filten = the warm blanket
• varma filtar = warm blankets

So varm is correct because filt is a common-gender singular indefinite noun.

In Swedish, when two singular count nouns are joined by och, it is normal to repeat the indefinite article if both nouns are separate items:

• en bok och en varm filt

That makes it clear we mean:

• one book
• and one warm blanket

If you said only en bok och varm filt, it would sound incomplete or unnatural in standard Swedish.

Med means with.

In this sentence, it most naturally describes the means or things used while comforting the child:

• She is trying to comfort the child with a book and a warm blanket

So the most likely interpretation is that the book and blanket are being used to comfort the child.

Like English, Swedish can sometimes be structurally ambiguous with with phrases, but here the natural reading is that med en bok och en varm filt belongs with the action trösta barnet.

Yes. Trösta is a very common Swedish verb meaning to comfort or to console.

Typical patterns:

• trösta någon = comfort someone
• Hon tröstar barnet = She comforts the child
• Han försöker trösta henne = He is trying to comfort her

So in your sentence, barnet is the direct object of trösta.

A rough guide for an English speaker:

• försöker ≈ fur-SUR-ker, but with Swedish ö sounds rather than plain English vowels
• trösta ≈ TRUR-sta, again with ö

The important thing is the letter ö, which does not have an exact English equivalent. It is somewhat similar to the vowel in British English bird or word, but rounded.

Also note:

• tr in Swedish is pronounced differently from English tr
• stress usually falls early: FÖR-sö-ker, TRÖS-ta

If you are learning pronunciation, ö is one of the key sounds to practice.