Translation:She had demonstrated that she can write well.
If you don't know the answer, I think it's better to use the hover to get the right answer, rather than form the wrong answer in your mind, only to have to mentally erase it and rewrite the correct answer.
Thing is, we all come back to these exercises several times during "strengthening" (as I am right now), and the next time it's easier to recall the right answer if you never previously formed the wrong one.
Too often, if you don't "cheat" (in a good sense of that word, as in using the hover), you may recall two different translations, one correct, one incorrect, and not be able to remember which is the right one.
Constant positive reinforcement is much better for learning languages than sporadic negative reinforcement.
I have learned to just always translate 'poder' as 'can' to keep the duo owl happy (except in the conditional or subjunctive, I suppose). In reality, it often should be translated as 'could', as in this sentence. In some other cases, 'can' and 'could' are interchangeable.
I used 'can' here, but it rejected my use of 'proven' and requires 'proved' here instead. Sigh.
I automatically (without thinking) used "could", because "could" is a past form of "can." I assumed (it seemed most logical to me, though I did not think about it) that the context was that she did the proving in the past.
DL accepted "could" (Nov., 2016)
I now see that the literal translation would be "can"; but I believe that , most likely, the context would be (if we had context) "she could write sell".
On a different note:
"She has shown that she can write well" is entirely appropriate English, under appropriate circumstances. Often a consistent history is a very coherent (logical) reason to believe someone "can" now do something.
My six year old nephew did not walk well four years ago. But he has proven to me that he can (now) walk well, and even run.
My driving record, which has had no accidents in 20 years, is reasonable grounds to believe that I have proven that I can drive well enough to avoid accidents. In terms of research design-- ongoing, continuous past history can give good reason to believe that the action can continue.
Absent any evidence to the contrary, only an unreasonable person would say I can not (am not able to) drive safely now. (I still do drive, and have driven almost every day, for the past two years,as well as the past 20 years, without accident, .) I am confident (and have reasonable grounds to believe) that I can drive home and to work, safely, tomorrow.
"I have demonstrated that I can drive safely." -- This is not incorrect English.
I don't know if 'he' is accepted, but, as a way to play with the ambiguity of Spanish sentences without specific pronouns, I always test what DL will accept and, 'She had shown that you (formal) can write well' is accepted as correct.
As no pronoun is provided for whom she is talking about, it could be 'she' (the seemingly logical answer), 'he', 'you (formal)' or even 'it' (who knows, she could be an AI programmer coding a system to pass the writing element of a Turing test or an ethologist training chimps in an ideographic script!). Just because we assume there is one obvious answer doesn't mean that others are necessarily incorrect.
'She had shown that he can write well' could be the description (as alluded to by another commenter) of how a teacher demonstrated a student's progress under her tutelage at a parent-teacher meeting, in which case 'Ella había demostrado que puede escribir bien' would be absolutely correct. After all, all DL sentences are without context and (in my opinion) using one's imagination can only aid in learning, not hinder it. But that's just my dos centavos.
Because it doesn't specifically say he. Obviously context plays a part, and colloquially when you're talking about multiple people it's common to just throw he and she into your sentences wherever you need them, but that's not necessarily correct. I believe that in this sort of instance, you would need to say "Ella había demostrado que su puede escribir bien". And even then, without context it would be unclear unless you replaced su with the actual name of the person. As it stands though, if you only mention one person, you assume that every other word is talking about that person as long as the word is in the correct form.
Again, not 100% sure on this though, so please feel free to correct me!
While it's not a correct translation, it is correct English grammar. The DL translation is bad English grammar because it mixes remove past with present tense in an illogical way.
However, I'm not sure at all if that is not what Spanish speakers do, so the sentence may be correct in Spanish. It's just the translation that's wrong, because one should sacrifice the literal if it creates a grammatical error in English, and should use good idiomatic English, even if it changes the tense.
I am not a native Spanish speaker but from several comments that I've seen from native speakers, it seems to be a general rule in Spanish that once the subject has been identified (in this case "ella" at the beginning of the sentence) it is assumed they will continue to be the subject of verbs until otherwise noted.