Identify the parent chain

The structure corresponds to a six-carbon chain containing both a double bond and a triple bond, so the parent hydrocarbon is hex-en-yne.

Number the chain to give lowest locants to multiple bonds

For compounds containing both $C=C$ and $C\equiv C$, numbering is chosen so that the set of locants for multiple bonds is lowest.

Possible numbering gives:

• From one end: double bond at $2$, triple bond at $4$
• From the other end: double bond at $4$, triple bond at $2$

Since in a tie the double bond gets the lower number, we choose: $\text{hex-2-en-4-yne}$

Locate the substituent

With this numbering, the bromo substituent is on carbon $2$.

So the name becomes: $2\text{-Bromo-hex-2-en-4-yne}$

Assign stereochemistry of the double bond

At the double bond, assign CIP priorities:

• On one carbon, $Br$ has higher priority than $CH_3$ / H-type substituent.
• On the other carbon, the carbon chain has higher priority than H.

The higher-priority groups are on opposite sides of the double bond, so configuration is E.

Hence the full name is: $\boxed{(2E)\text{-2-Bromo-hex-2-en-4-yne}}$

Check options

• A: $(4E)$-5-Bromo-hex-4-en-2-yne ❌
  • Multiple bonds not numbered correctly.
• B: $(2E)$-2-Bromo-hex-4-yn-2-ene ❌
  • Although similar in meaning, IUPAC format prefers en before yne in the suffix pattern as hex-2-en-4-yne.
• C: $(2E)$-2-Bromo-hex-2-en-4-yne ✅
• D: $(4E)$-5-Bromo-hex-2-en-4-yne ❌
  • Wrong substituent position and stereodescriptor numbering.

Final answer

The correct option is: $\boxed{\text{C}}$