After reading the IUPAC rules, I found the parent chain to be from the $\ce{CH2}$ numbered 1 to the one numbered 9, and I considered both methyl groups as branches. However, my confusion lies in naming the branch in the middle (depicted in red color) since it contains a double bond:
According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book), the traditional name ‘vinyl’ for $\ce{CH2=CH-{}}$ is no longer used for preferred IUPAC names and is only retained for general nomenclature. The preferred name is the systematic name ‘ethenyl’.
Therefore, the name for the compound that is given in the question ‘3,7-dimethyl-5-vinylnona-1,8-diene’ could be found in the literature; however, the preferred IUPAC name in accordance with current recommendations is 5-ethenyl-3,7-dimethylnona-1,8-diene.
