Both are passive and both are idiomatic. The version with the gerund-participle is what Huddleston & Pullum (2002) call a concealed passive construction and the version with the infinitive an ordinary passive construction:
With a small number of catenative verbs, notably need, require, deserve and want, a gerund-participial may be passive while ...
The answer is (4)to send- to hang
"Please do not forget to send me some pretty presents to hang on my tree."
It is a simple sentence of requesting/ordering someone to do something for you in a friendly way.
I think the answer is 4) because:
-Forget + doing sth: you did sth in the past, but you forget it
-Forget + to do sth : there is sth you will do in the future, but you don't remember that you must/ have to do it
I am afraid that, as the EngVid page on Gerunds and Infinitives listed by FumbleFingers says, this is simply an arbitrary rule of usage.
I disagree with a few of the statements on the page. but I agree with its general point nd with most of the specific classifications given there, and i can't advance any systematic criterion for which verbs take an ...