とても〜ない
Simply cannot, no matter what (absolute negation of possibility)machine translation
Formationとても + potential verb in the ない-form
とても normally means "very," but with a negative — especially a negative potential form — its meaning changes completely to "no matter how hard I try, I can't." This is a classic exam trap: とても〜ない does NOT mean "not very…" (that's あまり〜ない, which you learned at N5) but a flat-out denial of possibility.
Usage: place とても before a potential verb in the ない-form — とても解けない (there's no way I can solve it), とても信じられない (I simply can't believe it), とてもできない. The nuance is that the speaker has imagined it or tried it, then given up: "no, I just can't."
Distinguish: 全然〜ない negates the whole matter objectively, while とても〜ない emphasizes the speaker's helplessness before something beyond them. To say you "can't finish something completely," use 〜きれない (see the lesson on 〜きる／〜きれない).
Examples
こんなに難しい問題は、とても解けない。
A problem this hard — there's no way I can solve it.
彼の話はとても信じられない。
I simply can't believe his story.
一日でこの本を全部読むなんて、とてもできない。
Reading this whole book in one day? There's no way I could do that.