Hi Sasha, We object /do not object TO (something). We mind / do not mind (something). Often, this "something" is a *gerund or gerund phrase. Examples: Would you mind turning the volume of the radio down? Would you mind turning down the volume of the radio? Would you object to turning the volume of the radio down? Would you object to turning down the volume of the radio? *Gerunds function as nouns but take the form of verbs that have an -ing suffix. Personally, I find the use of "object to" overly formal if you are speaking to a friend or family member. However, if you are doing this as a grammar exercise, that's fine.

An upvote for Mr. Lance for an excellent answer. Here is some more information. 1 - "Would you mind to turn the radio down?" (incorrect) should be "Would you mind turning the radio down?" (correct). The verb "to mind something" takes a noun (or gerund) as a direct object. "To turn" is an infinitive and is incorrect here. "Turning" is a gerund and is correct here. 2 - "Would you object to turning the radio down?" (correct) The verb "to object to something" takes a noun (or gerund) as a prepositional object. "To turning" is a preposition and gerund.

You need to use continuous in this instance: 1) Would you mind turning the radio down 2) Would you object to turning the radio down