The first branch of carbon chain should have the lower locant - 2,4-dimethyl hexane (not 3,5-...), 3-ethyl-2,4-dimethyl hexane, 3-ethyl-2,5-dimethyl hexane, 3-ethyl-2,2,4-trimethyl hexane, 3-ethyl-2,4,5-trimethyl hexane.
4-Ethyl-2,2,3,3-tetramethyl hexane is the correct name.
Search for branched hydrocarbons nomenclature.