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enquire / enquiry
inquire / inquiry
In British English there's a subtle difference between enquiry and inquiry, but Americans tend to use inquiry for both, as a result the lines are getting blurred, but generally:-

If you enquire about someone or something you ask about them.

For example:-

She enquired about his health. He was touched by her enquiry..

In British English we use inquiry when we're talking about more official investigations.

For example:-

She thought the committee had no right to inquire into her politics.

There was an inquiry into the death of Ian Tomlinson.

!Note - If you can't decide, stick to inquire / inquiry. You'll sound like an American, but that can't be helped.