The related question and answer that were already linked in this question also cover this case. If the other answer is too long and confusing, we can limit it to the rules (taken from Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book)) that are relevant in this case:
P-44.1 SENIORITY ORDER FOR PARENT STRUCTURES
When there is a choice, the senior parent structure is chosen by applying the following criteria, in order, until a decision is reached. These criteria must always be applied before those applicable to rings and ring systems (see P-44.2) and to chains (see P-44.3). Then criteria applicable to both chains and rings or ring systems given in P-44.4 are considered.
P-44.3.2 The principal chain has the greater number of skeletal atoms [criterion (b) in P-44.3].
Two or more names may result based on the same senior parent structure selected according to P-44, because of different substitution patterns or multiple occurrences of the same senior parent structure. A parent structure is defined (P-15.1) as a parent hydride, for example, benzene, a functionalized parent hydride, for example cyclohexanol, or a functional parent compound, for example, acetic acid. Application of the criteria in this section will generate the preferred IUPAC name.
P-45.2.1 The preferred IUPAC name is based on the senior parent structure that has the maximum number of substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
P-45.2.2 The preferred IUPAC name is based on the senior parent structure that has the lower locant or set of locants for substituents cited as prefixes (other than ‘hydro/dehydro’) to the parent structure.
Therefore, for a correct name, only the longest chain is relevant for the choice of a principal chain in this case. Only if there is still a choice, substituent groups that are expressed as prefixes (e.g. ‘bromo-’) are taken into account for the preferred name according to P-45.2.1 and P-45.2.2.
Thus, the correct name for the compound that is given in the question is indeed 3-(bromomethyl)-3-methylpentane and not 1-bromo-2-ethyl-2-methylbutane since the pentane chain has the greater number of skeletal atoms.