Incorrect spelling, explanation: careing is the mistaken form of the adjective caring. In adjectives created from verbs with silent e at the end, e have to be removed before adding the suffix -ing. So, if caring comes from the verb care, when we apply the mentioned rule, we get the spelling caring as the correct one and see that careing is clearly a mistake.

Correct spelling, explanation: caring is the correct spelling of an adjective originating from the verb care. Because care ends with a silent e, to create an adjective we have to drop the letter e before adding the suffix -ing. For that reason, spelling careing is incorrect, because the only correct one is caring without e.

Definition of caring:
adjective: someone or something that is compassionate, kind, and concerned for others;
I have always considered her a caring person and her work with elders shows clearly how much empathy she has.
She is more caring towards her dog than her children, it is bizarre.

Incorrect spelling