Learn French: L'entreprise veut sincèrement compenser ce préjudice. - The company sincerely wants to compensate for this damage.

Yes, it is a major false friend! In French, un préjudice means damage, harm, or a loss (often used in legal or business contexts). If you want to say 'a prejudice' (a preconceived opinion) in French, the correct word is un préjugé.

Why is there no preposition after compenser? English uses 'compensate for'.

In French, compenser is a direct transitive verb, meaning it connects directly to its object without a preposition. You simply say compenser quelque chose, which is why there is no pour before ce préjudice.

Why is sincèrement placed after veut instead of before it like the English 'sincerely wants'?

In French, adverbs that modify a conjugated verb are placed immediately after that verb. It is grammatically incorrect in French to put an adverb between the subject and the verb, so you must say veut sincèrement rather than sincèrement veut.

This lesson is about quoi que + subjunctive. How could we add 'whatever it costs' to this sentence to practice that?

You could add the common set phrase quoi qu'il en coûte (whatever it costs / at all costs) to the end of the sentence. This uses quoi que followed by the subjunctive of coûter. The full sentence would be: L'entreprise veut sincèrement compenser ce préjudice, quoi qu'il en coûte.