V-(으)시겠어요?

Would you (mind/like to)...?, Why not...?

Structure
Vowel-final: 가다 → 가 + 시 + 겠어요? → 가시겠어요? (would you go?)
Consonant-final: 먹다 → 먹 + 으시 + 겠어요? → 먹으시겠어요? (would you eat?)
ㄹ-final: 살다 → 살 − ㄹ + 시 + 겠어요? → 사시겠어요? (would you live?)
Explanation

V-(으)시겠어요? is used when politely suggesting something to the listener or when asking about the listener's preference or intention.

It corresponds to 'Would you (mind/like to)...?' or 'Why not...?' in English and conveys a much more formal and polite feeling than -(으)ㄹ래요?/-(으)실래요?.

When the stem ends in a vowel, -시겠어요? is used, and when it ends in a consonant, -(으)시겠어요? is used.

Details