Correct spelling, explanation: the form comes from Latin that consisted of word-forming element inter meaning between. It was also used extensively as a prefix. The word interest was derived from the verb interesse where esse meant to be. Considering the origin of the word, we can conclude that only the form interest is correct, intrest is a typo where the vowel e between the consonants t and r is missing. Therefore, interest is wrong.

Definition of interest:
noun/verb, when someone has an interest in something, it gives a lot of attention to it,
His interests are cycling and cooking.
He has never seemed to show an interest in astronomy.

Phrases with interest:
have one’s best interest at heart – to be concerned with what would benefit someone,
Although you don’t agree with my decisions, I have your best interests at heart.
in the interests of justice – to be just or fair,
In the interest of justice, you shouldn’t leave the country.

Incorrect spelling, explanation: intrest is a misspelling or a typo probably because of the pronunciation of this word as /ˈɪntrest/. The word has its roots in Latin as interest where inter functioned as a prefix. It clearly shows that only interest is accurate, the spelling intrest is incorrect.

Incorrect spelling

Incorrect spelling

Incorrect spelling