Had better is an idiomatic verb phrase meaning “ought to, must.” It resembles an auxiliary verb in that its form never changes to show person or tense and that it can’t follow another verb in a phrase (that is, you can’t say He will had better leave, for example).
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When speaking, people have a tendency to leave out had: You better clean up your room! But in writing, you had better keep had, either in full or as a contraction: You had better not do that or You’d better not do that.
