The infinitive in the sentence My mother has finally learned to ride my brother’s motorcycle is to ride and functions as a A) noun.
An infinitive is the basic form a verb preceded by the particle to. Infinitives can act as nouns, adjectives or adverbs. As a noun, an infinitive can appear either as the subject of the sentence or as the direct object, like in this sentence, because it is the thing -noun or pronoun- that receives the action expressed in the verb. In this case, learned is the action and to ride… is the direct object, so it is acting as a noun.