Identify the principal functional group

The compound contains a carboxylic acid group, $-COOH$.

When a $-COOH$ group is attached to a ring and the carbon of the carboxyl group is not part of the ring, the correct IUPAC form is: $\text{cycloalkane carboxylic acid}$ So the parent name must be of the form: $\text{cyclopentane carboxylic acid}$

Hence, options with cyclopentanoic acid are incorrect.

Therefore, B and D are eliminated.

Fix the numbering of the ring

In cycloalkane carboxylic acids, the ring carbon attached to the $-COOH$ group is numbered as C-1.

Now number the ring in the direction that gives the substituents the lowest set of locants.

Locate the substituents

From the given structure, the substituents on the cyclopentane ring are:

• one methyl group
• one bromo group

Taking the carbon attached to $-COOH$ as C-1, numbering in the correct direction gives:

• methyl at C-2
• bromo at C-4

So the name becomes: $\boxed{4\text{-Bromo-}2\text{-methylcyclopentane carboxylic acid}}$

Check with options

• A: $3$-Bromo-$5$-methylcyclopentane carboxylic acid $\rightarrow$ incorrect locants
• B: $3$-Bromo-$5$-methylcyclopentanoic acid $\rightarrow$ wrong parent suffix
• C: $4$-Bromo-$2$-methylcyclopentane carboxylic acid $\rightarrow$ correct
• D: $5$-Bromo-$3$-methylcyclopentanoic acid $\rightarrow$ wrong parent suffix

Final answer

$\boxed{\text{C: 4-Bromo-2-methylcyclopentane carboxylic acid}}$

Comparison with stored correct answer

Stored correct answer: C

My derived answer: C

Hence, they agree.