"Hans is Karl's brother."
Translation:Hans ist der Bruder von Karl.
Ah, sorry, I just missed the "a". I'm not sure about that. "Hans ist Karls Bruder" (just like "Hans is Karl's brother") makes me think that Karl has only one brother, while "ein Bruder" would be only one of several brothers.
Apart from that, "Hans ist ein Bruder von Karl" is a correct German sentence.
vom is a contraction of von dem, i.e. "of the".
In standard German, we don't use the definite article with names or people -- we don't talk about "the Karl". (You may hear the definite article in informal language or in dialects, though, as in Der Hans ist der Bruder vom Karl. But that's not accepted on this course.)
vor is a preposition meaning "before" or "in front of". It's not a contraction of von der, in case you though that -- von der does not have a contraction in standard German. (In my colloquial speech, it does have one, but that's vonner, not vor.)
So it's just "the brother of Karl" der Bruder von Karl, not "the brother of the Karl" or "the brother in front of Karl".
Is genitive possible?
Yes - Hans ist Karls Bruder is also possible.
As in "Hans ist der Bruder des Karles"
No. We don't use the the definite article with people's names in standard German (though that's common in some dialects and also colloquially in some regions) -- muhc like how we don't talk about "the Karl" in English. So des Karls "of the Karl" is wrong.
der Bruder Karls would be theoretically possible, but in practice, people would say Karls Bruder instead.
sein (to be) is not a transitive verb that takes a direct object in the accusative case.
(Which is also why you can't turn it into the passive voice: "Karl's brother is been by Hans"(???).)
Instead, "to be" is a copula or linking verb that links a subject to a predicate that says something about the subject.
Such predicates are in the nominative case in German.
Same with werden (to become), e.g. er ist ein guter Vater geworden "he has become a good father".
der Bruder von Karl is one noun phrase -- not two separate ones.
Also, "dative noun" and "accusative noun" work in that order when you're referring to indirect and direct objects of a verb.
Karl is not an object of the verb; it's part of a prepositional phrase headed by von. The fact that von requires the dative case is irrelevant for the positioning in the sentence.
why is "Hans ist Karls Bruder" marked as wrong?
Hans ist Karls Bruder. is one of the accepted translations. Perhaps you made a small mistake somewhere? Can you show us a screenshot of your rejected answer, please, by uploading it to a website somewhere and telling us the URL?
Would you not expect to hear "Hans ist Karls Bruder" in casual conversation?
Could go either way; I wouldn't be surprised to hear it like that.
As with English, it's more likely to use it with people (we say "the roof of the house" with inanimate house but "my brother's name" with animate "my brother", rather than "the name of my brother").
What definitely sounds extremely old-fashioned is using the genitive of a noun before another noun, e.g. Sie ist meines Vaters Schwester for "She is my father's sister" -- I would expect Sie ist die Schwester meines Vaters or Sie ist die Schwester von meinem Vater.
You used the definite article des before Karl -- we don't use the definite article before names in standard German. (Though a number of dialects do that.)
So you would write Karl sieht Julia and not der Karl sieht die Julia, and you wouldn't write des Karl.
If you want to use the genitive, the usual order is Karls Bruder, not der Bruder Karls.
(And note the -s here, which appears not only on male names but also female names, e.g. Julias Bruder; der Bruder Julias.)
Karls Bruder is accepted here; the less common der Bruder Karls is not.