Identify the longest chain containing the double bond

We must choose the parent chain such that:

• it contains the double bond,
• it is the longest possible chain.

From the given structure, the correct parent chain has 6 carbon atoms and includes the double bond.

Therefore, the parent hydrocarbon is hexene.

Number the chain to give the double bond the lowest locant

In IUPAC nomenclature, the double bond gets priority in numbering.

Numbering from the end nearer the double bond gives: $\text{hex-2-ene}$ and not hex-4-ene, because: $2 < 4$

Locate the substituents

After numbering the main chain as hex-2-ene, the substituents are:

• one ethyl group at carbon 4,
• one methyl group at carbon 3.

So the substituent positions are:

• $4$-ethyl
• $3$-methyl

Arrange substituents alphabetically

In IUPAC naming, substituents are written in alphabetical order:

• ethyl
• methyl

Hence the name becomes: $4\text{-ethyl-}3\text{-methylhex-2-ene}$

Check options

• A: 4-methyl-3-ethylhex-4-ene
  Incorrect: double bond locant is not lowest.

• B: 3-ethyl-4-methylhex-4-ene
  Incorrect: double bond locant is not lowest.

• C: 4-ethyl-3-methylhex-2-ene
  Correct.

• D: 4,4-diethyl-3-methylbut-2-ene
  Incorrect: does not use the longest valid parent chain.

Final answer

The correct IUPAC name is: $\boxed{4\text{-ethyl-}3\text{-methylhex-2-ene}}$