Det ser ud til, at hun har glemt sin notatbog derhjemme, så jeg tager en ekstra med til hende.

Det ser ud til, at ... is a very common Danish way to say It looks like ... or It seems that ....

Literally, the parts are:

• det = it
• ser ud til = looks/seems
• at = that

So:

• Det ser ud til, at hun har glemt sin notatbog ... = It looks like she has forgotten her notebook ...

This is a fixed expression, so it is best to learn ser ud til together as one unit.

Har glemt is the present perfect, like English has forgotten.

• hun har glemt = she has forgotten
• hun glemte = she forgot

In this sentence, Danish uses the perfect because it describes a past action with a present result: she forgot it, and now the notebook is still at home.

So har glemt feels very natural here.

This is a very important Danish point.

Danish uses sin/sit/sine when the possessor is the subject of the same clause.

Here, the subject is hun and the notebook belongs to her, so Danish uses the reflexive possessive:

• hun har glemt sin notatbog = she has forgotten her own notebook

If you said hendes notatbog, it would usually suggest someone else’s notebook belonging to her/that woman, not her own in the reflexive sense.

So:

• Hun tog sin jakke. = She took her own jacket.
• Hun tog hendes jakke. = She took her jacket, meaning another woman’s jacket.

The form depends on the noun that follows:

• sin
  • common gender singular
• sit
  • neuter singular
• sine
  • plural

notatbog is a common gender singular noun, so the correct form is sin.

Examples:

• sin bog = her/his own book
• sit hus = her/his own house
• sine bøger = her/his own books

Derhjemme means at home, often with a slight sense of back at home / at her home / at home where she lives.

In many situations, derhjemme and hjemme are both possible, but derhjemme can feel a bit more specific or pointed.

So:

• hun har glemt sin notatbog derhjemme = she has forgotten her notebook at home

It helps emphasize the location where it was left.

Standard Danish punctuation usually places commas around subordinate clauses.

Here:

• Det ser ud til, at hun har glemt sin notatbog derhjemme has a subordinate clause beginning with at
• then så introduces the next main clause: så jeg tager en ekstra med til hende

So the commas help show the structure clearly.

Very roughly:

• main clause: Det ser ud til
• subordinate clause: at hun har glemt sin notatbog derhjemme
• new main clause: så jeg tager en ekstra med til hende

Both are possible in Danish, but they are not exactly the same in structure.

In the sentence you were given, så works like so, connecting two clauses:

• ..., så jeg tager en ekstra med til hende. = ..., so I’m taking an extra one for her.

Here jeg comes before tager because this is a normal clause with the subject first.

But Danish also has V2 word order in main clauses, so in some contexts you can get:

• ..., så tager jeg en ekstra med til hende.

That version can sound a little more like so then I’ll take an extra one or give slightly more emphasis to the action.

In everyday language, both may occur, but så jeg tager ... is perfectly natural as a connected so-clause.

Danish often uses the present tense to talk about a planned or intended future action, just like English can do.

So:

• jeg tager en ekstra med til hende literally = I take an extra one with me for her but naturally = I’ll bring an extra one for her / I’m bringing an extra one for her

This is very common when the speaker has already decided what to do.

En ekstra means an extra one.

The noun notatbog is understood from the earlier part of the sentence, so Danish leaves it out.

This is very common:

• Jeg tager en ekstra med. = I’ll bring an extra one.

The missing noun is recovered from context. Here it means an extra notebook.

Here med means something like along / with me.

So:

• tage noget med = bring something along
• jeg tager en ekstra med = I’ll bring an extra one along

Without med, tage just means take. With med, it often means take along / bring depending on perspective.

That is why the full phrase naturally translates as I’ll bring an extra one for her.

Because Danish uses til for the idea of for/to someone in this kind of sentence.

• til hende = for her / to her

So:

• jeg tager en ekstra med til hende = I’m bringing an extra one for her

You cannot normally drop til here.

Yes, notatbog is a normal word meaning notebook.

It is a compound noun:

• notat = note
• bog = book

So literally it is a note-book.

Like many Danish compounds, it is written as one word:

• notatbog = notebook
• not notat bog

Yes:

• Det = it
• ser ud til = looks like / seems
• at = that
• hun = she
• har glemt = has forgotten
• sin = her own
• notatbog = notebook
• derhjemme = at home
• så = so
• jeg = I
• tager = take / am taking / will take
• en ekstra = an extra one
• med = along / with me
• til hende = for her

So the sentence structure is:

It looks like she has forgotten her notebook at home, so I’m bringing an extra one for her.