"He has asked me to write a book."
Translation:Hij heeft me gevraagd een boek te schrijven.
Are you asking me, or did you write your comment for the moderators?
The various units in a DL course focus on particular topics. But I've noticed that not every sentence in a particular unit always exemplifies the topic.
Sometimes that may be a mistake by the DL computer program or the language team, but sometimes the odd sentence may be there deliberately to keep us on our toes.
Is that such a bad thing?
There are two clauses/phrases in the DL sentence here:
1. Hij heeft me gevraagd ...
2. ... een boek te schrijven.
So the participle "gevraagd" DOES come at the end of its clause/phrase.
The second clause/phrase here is an infinitive phrase that includes the infinitive "te schrijven" and the object of that infinitive, "een boek".
In English we say "to write a book", but in Dutch the object precedes the infinitive.