Take a sentence such as Я имею основания ей не верить.
If основания is accusative, then it must be plural. And thus the translation will be: I have reasons not to trust her.
But if основания is genitive, then it must be singular. And thus the translation will be: I have a reason not to trust her.
I'm sure native Russian speakers can identify the correct form instantly from years of input, but what is going on behind the scenes here?
Maybe I must simply know that there are no rules that make основания genitive here, and therefore it must be accusative, and mean reasons, not reason?