Hanaseru

Konnichiwa!

hanaseru is the informal "can" form of the verb to speak. The rule for this verb is to replace the ending: -u with -eru.
So it would be something like this: hanasu - to speak and hanaseru - can speak.
Example: boku wa nihongo o hanaseru - I can speak Japanese

Now, you may know the form like this: hanaseMASU, which is the polite form of hanaseru.
For a better explanation, find a post made by me, named: <<"Can you ?" questions>>, where Sayaka-sensei explained these things way better than me (I'm not good at explaining things, gomen).

Hope this was of help !

コッド

Here's a copy of the post Coddo-san mentioned:

U-ending verbs:
tsukau "to use" > drop the 'u' and add 'eru' > tsuakaeru "can use"
iku "to go" > drop the 'u' and add 'eru' > ikeru "can go"
hanasu "to speak/talk" > drop the 'u' and add 'eru' > hanaseru "can speak/talk"

To form the polite versions of these three "CAN" verbs (tsukaeru, ikeru, hanaseru) you need to replace -RU with -MASU, so you get tsukaemasu, ikemasu, hanasemasu.

And, to make these questions, we just add the particle -KA > tsuakemasu ka? ikemasu ka? hanasemasu ka? "Can you use?" "Can you go?" "Can you speak?*

Now, taberu or "to eat" is a -RU ending verb so to form the CAN-verb we need to follow these rules:
taberu "to eat" > drop the 'ru' and add 'rareru' > taberareru "can eat"

To form the polite version of taberareru "Can eat" we need to replace -RU with -MASU - which gives us taberaremasu "I can eat".

To make this a CAN YOU? question we add the question particle KA > taberaremasu ka? "Can you eat?"