1. Would you prefer me to stay?
2. Would you prefer me staying?
Q1) Are both 1 and 2 correct English?
Q2) Do both 1 and 2 mean the same thing?
Is sentence 2 correct English? I'm not really sure about whether 2 is correct.
I'm sure 1 is perfectly correct since I have only seen the form "pefer someone to do" used.
fire1 1. Would you prefer me to stay? 2.
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fire11. Would you prefer me to stay?
2. Would you prefer me staying?
In the meaning that is normally intended, (1) is the "proper" form, and (2) is a slightly looser form that people may use colloquially.