Correct spelling, explanation: believe is a verb that has a long history. It comes from (among others) Middle English beleven, bileven, and from Old English belīefan (meaning to believe). The form beleeve was common till the 17th century, the spelling then changed, perhaps by the influence of relieve, etc. Right now, believe is the only correct form and we should remember that. You shouldn’t spell this verb as beleave.

Definition of believe:
verb, accept that (something) is true, especially without proof
I believe in God, so I go to church every Sunday.
When she told me about her trauma, I believed her every word.

Collocations, expressions and idioms with believe:
Some most commonly used collocations, expressions, and idioms include: can’t believe one’s own eyes, believe me, not believe a word of it, and believe it or not.
I see this demonstration, but I can’t believe my eyes.
Believe me, I don’t want to be her either.
I don’t believe a word you’re saying – Derek wouldn’t beat his friend!
Believe it or not, I’m a fan of country music.

Incorrect spelling, explanation: beleave is a mistake, but it can often occur because of two reasons. Firstly, the pronunciation. In this case, -lieve and -leave sound exactly the same. Secondly, the history. One of the old versions of this word is an equivalent to be- +‎ leave (meaning to give leave or permission to, permit, allow, grant). However, you should bear in mind that today’s shape of it is different. Believe is the only correct spelling.

Incorrect spelling

Incorrect spelling

Incorrect spelling

Incorrect spelling

Incorrect spelling

Incorrect spelling