The correct answer is "I've learned to get along with my brother ever since my father left the family business to him and me (B)."
All of the options are personal pronouns. There are two groups of pronouns nominative and objective. The nominative pronouns are used with subjects and are I, he, she, we and they. The objective pronouns are used as objects and are me, her, him, us, and them. You and its can be nominative or objective. It is important to note that when using more than one personal pronoun in a sentence they must be of the same type, both nominative or both objective. It is not correct to use one nominative and one objective personal pronoun for example. This eliminates him and I (C).
He and myself (D), includes a reflexive pronoun. A reflexive pronoun is a special kind of pronoun that is used when the object is the same as the subject. These include words like myself, yourself, herself, etc...In this sentence we cannot use a reflexive pronoun because the father left the business to his children. The subject and object are different.
This leaves us with two options he and I (A) or him and me (B). In order to determine which one is correct it is helpful to read the sentence with each of the personal pronouns separately. For example, "I've learned to get along with my brother ever since my father left the family business to he." With a basic understanding of the English language, we know this does not sound right. Therefore, it eliminates option A.