Nomenclature of an organic compound with benzene and double bonds
Since the compound that is given in the question doesn’t have a principal characteristic group (that would be expressed as a suffix and that would determine the senior parent structure), the ring has seniority over the chain when selecting the preferred IUPAC name. In general nomenclature and depending on the context, however, the chain may be favoured to recognize the unsaturated acyclic structure.

The corresponding subsection in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:


  P-44.1.2.2 Systems composed of rings and chains (exclusive of linear phanes)
  
  Two methods are recognized to name systems composed of rings and chains
  (exclusive of linear phanes).
  
  (1) Within the same class, a ring or ring system has seniority over a chain. When a ring and a chain contain the same senior element, the ring is chosen as parent. Rings and chains are chosen regardless of their degree of hydrogenation. As a consequence, this approach prefers the choice of a ring over a chain in systems composed of cyclic and acyclic hydrocarbons.
  
  (2) The context may favor the ring or the chain, so that, for example, substituents may be treated alike or an unsaturated acyclic structure may be recognized, or the one chosen has the greater number of skeletal atoms in the ring or in the principal chain of the acyclic structure.
  
  (…) For selection of a preferred IUPAC name, see P-52.2.8.


 


  P-52.2.8 Selection between a ring and a chain as parent hydride
  
  Within the same heteroatom class and for the same number of characteristic groups cited as the principal characteristic group, a ring is always selected as the parent hydride to construct a preferred IUPAC name. In general nomenclature, a ring or a chain can be the parent hydride (see P-44.1.2.2).


Method (1): (penta-3,4-dien-2-yl)benzene (preferred IUPAC name; ring preferred to chain)

Method (2): 4-phenylpenta-1,2-diene (unsaturated acyclic structure may be recognized)

Therefore, the preferred IUPAC name for the compound that is given in the question is (penta-3,4-dien-2-yl)benzene.



Note that parentheses are used around substituent prefixes (here: “penta-3,4-dien-2-yl”) to separate locants referring to different structural elements.

Also note that locants are placed immediately before that part of the name to which they relate (i.e. “penta-1,2-diene” and not “1,2-pentadiene”).