Step 1: Analyze the Structure.
The given structure is:
\[
\text{CH}_3
\]
\[
\text{CH}_3-\text{C}-\text{C}-\text{C}-\text{C}-\text{CH}_3
\]
\[
\quad \mid \quad \mid
\]
\[
\text{CH}_3 \quad \text{CH}_2\text{CH}_3
\]
This is a hexane derivative with substituents on the carbon chain.
Step 2: Identify the Parent Chain.
1. The longest continuous carbon chain has 6 carbon atoms, so the parent hydrocarbon is hexane.
2. The substituents are:
A methyl group (\(--\mathrm{CH}_3\)) at the 2\textsuperscript{nd} carbon.
An ethyl group (\(--\mathrm{CH}_2\mathrm{CH}_3\)) at the 4\textsuperscript{th} carbon.
Step 3: Number the Carbon Chain.
Start numbering the carbon chain from the end that gives the lowest possible numbers to the substituents.
If we start from the left, the substituents are at positions 2 and 4:
Methyl group at C-2.
Ethyl group at C-4.
Step 4: Write the IUPAC Name.
The IUPAC name follows the format: substituent position-number-substituent name-parent name.
The substituents are listed in alphabetical order:
"ethyl" comes before "methyl".
Therefore, the name is:
\[
4\text{-Ethyl-2-methylhexane}
\]
Step 5: Verify the Options.
Option 1: \( 2\text{-Methyl-4-ethylhexane} \)
Incorrect: The ethyl group should come before the methyl group alphabetically.
Option 2: \( 4\text{-Ethyl-2-methylhexane} \)
Correct: Follows the correct alphabetical order and numbering.
Option 3: \( 5\text{-Methyl-3-ethylhexane} \)
Incorrect: The numbering starts from the wrong end, leading to incorrect positions.
Option 4: \( 3\text{-Ethyl-5-methylhexane} \)
Incorrect: The numbering starts from the wrong end, leading to incorrect positions.
Final Answer: \( \boxed{4\text{-Ethyl-2-methylhexane}} \)