I'm sorry I couldn't be more specific in my question title. I don't have the vocabulary to pinpoint what exactly I have difficulties with yet; therefore, I'll try to explain through examples.

I came across this exercise on College Panda and was confused:

"Neither the employees nor the owner (cares/care) about the customer."

I thought the answer was "care" because "employees" and "owner" could be summarized with the pronoun 'they'.

And you would say, "They(plural) don't care about the customer".

But apparently, the answer is, "Neither the employees nor the owner cares about the customer." Does that mean, when using two nouns in tandem, you are supposed to base the subject/verb agreement on the noun directly preceding the verb?

E.g. with "My daughters and my son play/plays together", the verb "play" must agree with "son"; with "My son and daughters play/plays together", the verb "play" must agree with "daughters"?

Sorry about any confusion.

TL;DR: Please explain why the answer isn't "care"

Thanks in advance!