As a noun regret is emotional pain on account of something done or experienced in the past, with a wish that it had been different; a looking back with dissatisfaction or with longing.
As an adjective remorseful is (of a person) feeling or filled with remorse.
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This is a catenative verb that takes the gerund (the (-ing) form), except in set phrases with tell, say, and inform, where the to infinitive is used. See
(en adjective )