I forgot my homework at home is right:lol:
Well, that's settled then.
I agree that it should be 'left'.
I agree that 'left' does not imply intention.
I would say that 'forgot' is perfectly acceptable colloquial AE.
I've had this discussion with many many English teaching friends over here from the US, UK, Canada, Oz and NZ and generally they agree.
So there you go... it's many many vs emergine. I love science!
BTW, as a newb forgive me but I couldn't find an explanation of what these polls are for: acceptable standard usage (including vernacular and colloquialisms) or acceptable for teaching...?! I teach that 'forgot' in this context is wrong, but acceptable conversational colloquial AE.
BTWII
I forgot my homework at home. SVO+adjunct modifier
...
Since language is not a science, the fact that you can qualify a sentence with a grammatical 'formula' like this does not necessarily give it any more weight.
I left my homework. (Not OK)
Similarly, contextually, they may be occasions when this is acceptable.