"She has no children of her own."
Translation:Hon har inga egna barn.
egen is an adjective and those can go in two places: either before the noun they describe like here, or after the verb if there's a verb like är.
So either en röd bok 'a red book' or boken är röd 'the book is red'.
Here of course we have the first case. In English, you don't use an adjective for this word, but instead prefer a prepositional phrase, 'of her own' which goes after the noun. If the sentence had said something else, like Hon har inga gamla barn 'She has no old children' it would have worked the same way in both languages.
why is it "inga" instead of "inte"? It seems to me like the "inte" could go with the verb--like "doesn't have". Is it because the English sentence is "has no" instead of "doesn't have"? They do mean the same thing.
Yes. You're asked to translate "no children" and thus you'll have to translate to "inga barn".
It is, but inga is used with plurals, as it egna. Had it been just one child, it'd been "inget eget barn".