This pattern conveys a strong, often rhetorical desire for a solution or outcome that seems unlikely. It is the negative counterpart of 'ものか' (which expresses determination) but here it is a question form that implies 'is there no way...?' or 'can't it be that...?' The nuance is emotive and almost pleading, as if the speaker is grasping at hope despite doubts. It is used in both spoken and written Japanese, but it is somewhat literary or formal in tone, especially with だろうか. In casual speech, you might hear it in dramas or emotional appeals, but it is not everyday colloquial. The pattern often appears with verbs like ある (exist), できる (can do), or other state/possibility verbs, and it frequently co-occurs with 何とか (somehow) or どうにか (by some means). For example, 何とかならないものか means 'Can't something be done about it?' The key is that the speaker is not simply asking a question; they are expressing a heartfelt wish, often with a tinge of frustration or resignation. Note that ものか without the question particle (i.e., as a statement) means 'never' or 'no way' (e.g., 負けるものか = I'll never lose), but adding the question か turns it into this hopeful question. So the difference is crucial: the question form changes the meaning entirely.