Step-by-step explanation:
In the sentence "We'd love to know with you," the phrase "to know" is (C) an Infinitive.
Infinitives are verb forms that are typically preceded by the particle "to." In this case, "to know" functions as the complement of the verb "love" and expresses the action or intention of wanting to know something. It functions as a noun in this sentence, serving as the direct object of the verb "love."