Often, if you need to specify the agent for a passive construction, you're probably better off using the active voice. I would prefer "Can we solve this problem?" and "Have you shut the door?" to either passive. Use the passive (without an agent) where the emphasis is on the action, not the person(s) performing the action.
The active voice is speech that places the subject first, followed by a verb, and a place/object. Therefore the correct form is subject -- verb -- object.
The passive voice is speech that places the object first, followed by a verb (tense and a gerund), and the subject. Therefore the correct form is object -- verb -- subject.
None of these sentences are in the passive voice, since I (the subject), always comes first in each of them.
Most of these can be converted from active voice to passive voice:
I saw a person repairing my car. The car was being repaired by someone.
I saw a person repair my car. The car was repaired by someone.
I saw a car being repaired. / I saw a car be repaired. A car was being repaired.
I watched a man solve the problem. The problem was solved by a man.
I watched a man solving the problem. The problem was being solved by a man.
I watched a problem being solved. / I watched a problem be solved. The problem was being solved.
Notice that none of these use I, mainly because I couldn't find a logical incorporation. I suppose you could say (and use as a guideline):
The car was being repaired by someone I was watching.
The problem was solved as I watched.
The sentences I saw a car repaired and I saw a problem solved don't make sense to me because the past participle is just by itself. I would include words to tie it together with everything else:
I saw a car that was repaired.
I saw a car that had been repaired.
I saw a problem that was solved.
I saw a problem that had been solved.
I also think that adding a comma, though strange, is acceptable:
I saw a car, repaired.
I saw a problem, solved.
Best Answer
Grammar checkers cannot know when the passive voice is appropriate or not. In many cases, they can't even identify the passive correctly. You should always treat such advice skeptically.
Even worse, simple-minded style guides like The Elements of Style have convinced many writers and educators that the passive voice is always bad, when of course that is false. So advice from native speakers about the passive voice cannot always be trusted.
You seem to have identified the appropriate guideline. If the agent of solve is irrelevant to the context or cannot be known, then the passive might be the best choice.
Furthermore, if the topic of the passage is problem, then a structure that makes problem the grammatical subject is probably a good choice.