Identify the principal functional group

The compound contains both:

• a carboxylic acid group, and
• an ester group of the form $-COOCH_3$ (methoxycarbonyl substituent).

By IUPAC priority, carboxylic acid has higher priority than ester. Therefore, the parent name must end with ...oic acid.

Choose the parent carbon chain

The longest chain containing the carboxylic acid carbon has 6 carbons. Hence the parent is: $\text{hexanoic acid}$

Number the chain

Numbering starts from the carboxylic acid carbon as carbon 1.

Then the substituents are located as:

• a methyl group at carbon 2,
• a methoxycarbonyl group at carbon 5.

Write the substituents in the name

So the name becomes: $\text{5-methoxycarbonyl-2-methylhexanoic acid}$

Check the options

• A: 5-Methoxycarbonly-2-methylhexanoic acid
  This clearly intends 5-methoxycarbonyl-2-methylhexanoic acid. Spelling typo aside, this matches the correct IUPAC name.

• B: Methyl-5-carboxy-2-methylhexanoate
  Incorrect, because ester is taken as principal group instead of carboxylic acid.

• C: 2-Carboxy-5-methoxycarbonylhexane
  Incorrect, because carboxylic acid should be named as suffix, not just as a substituent.

• D: Methyl-6-carboxy-2,5-dimethylhexanoate
  Incorrect functional group choice and incorrect substituent pattern.

Final answer

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

Comparison with stored answer

Stored correct answer = A.
My derived answer = A.
Therefore, they agree.