For "Your (singular and formal)", read "Your (formal)" - formal Sie can be singular or plural.
How do you know that the other answers are not accepted, though? Did you try one of them and get it rejected? If so, I hope you've reported it so that course maintainers will see the suggestion? (Posting here is less likely to be useful.)
If it's not accepted, then the only reason I can think of is that it's a forgotten alternative, and I would encourage you to report it as "my translation should be accepted" the next time you come across this sentence and it marks you wrong.
Do check whether all the rest of the translation is correct, though, and you didn't inadvertently make a real mistake somewhere else in the sentence.
And without context, it's simply ambiguous.
Both versions will generally be accepted in Duolingo translations as in a single sentence without context, you can't tell the difference.
A bit like "How can you tell when 'you' refers to one person and when it refers to many people?" in English.
ihre is appropriate for a feminine singular noun (ihre Blume, ihre Gabel, ihre Tochter), but also for all plural nouns (ihre Söhne, ihre Välter, ihre Kinder, ihre Bücher).
ihr Kind is "her child" (one of them) and ihre Kinder is "her children" (lots of them).
A bit like how it's das Kind for "the child" but die Kinder for "the children" -- all nouns take the article die in the plural, and we don't have *das Kinder.
den is masculine accusative or plural dative.
So you would have, for example, Ich esse den Apfel. "I am eating the apple", because Apfel is masculine and the direct object of a verb is in the accusative case.
Or mit den Lehrern "with the teachers", because "teachers" is plural and mit requires the dative case.
ihrer and seiner with -er would be feminine genitive, feminine dative, or plural genitive -- der Name ihrer Katze "the name of her cat", das Haus seiner Tiere "the house of his animals", mit meiner Tochter "with my daughter".
ihres, seines are masculine genitive or neuter genitive: wegen ihres Sohnes "because of her son", die Mutter seines Kindes "the mother of his child"
For a sentence such as this one, how do you know if Ihre is for her or your (formal) in this context?
Only from context.
If this simple sentence was spoken by itself by someone, would you have to ask them for clarification?
But fortunately, conversations usually have context, and personal pronouns such as "you" or possessive determiners such as "her" usually refer to something that has been previously discussed and so it's less ambiguous what is being meant.
I'm afraid that's German for you.
Lowercase sie can mean "she" or "they", and uppercase Sie is for polite "you".
And then ihr (ihre, ihren, ...) can mean either "her" or "their", and uppercase Ihr (Ihre, Ihren, ...) for polite "your".
It probably doesn't help that ihr can also mean "you" (plural, informal), i.e. as a personal pronoun rather than as a possessive adjective.
About the closest English comes to such shenanigans is using "her" for both personal pronoun ("I see her") and possessive adjective ("Her bag is blue"), or "his" for both possessive adjective ("His shoes are blue") and possessive pronoun ("His are blue"), though all other persons keep those two separate (e.g. "my/mine, your/yours").
You have to relax & when you feel overwhelmed, go back & practice the earlier lessons! Thats what i do with Spanish mostly. Then SLOWLY work your way back to where you are now. It WILL help! And the bear reference is a rhyme. The BEAR drinks BEER. There's a difference...