core

If I am writing like this:-

I am frustrating to the core. First is it correct? Second here Does 'to the core' show that
I am frustrated enough from inside?

porwalrox said:

First is it correct?

I am afraid not.
'Frustrate' means 'prevent' or 'defeat' (somebody's aim or action).

For example, if you are trying to open the door, and I am holding it shut from the other side, then I am frustrating your efforts.

You could then say: 'I find this frustrating' or 'I am frustrated by this'.
This means: 'I find that this situation is frustrating me'.
In other words, your experience is that the situation is defeating your aim.

"To the core" means thoroughly. "He's evil to the core." "The sight of all those dead bodies shook me to the core." You can't really use it to say you're frustrated. It's a bit too strong for that.

Generally, frustration is an external phenomenon, so it would seem a little odd to speak of being 'frustrated to the core', as the 'core' (the heart, the very centre) is essentially internal.
Maybe, in extremis, I might say 'I was frustrated from pillar to post'.

Thanks @rhiragawr.