1. Identify the principal functional group

   The compound contains a carboxylic acid group, so the parent name must end with -oic acid.

2. Choose the longest chain containing the carboxyl carbon

   The longest carbon chain including the $-COOH$ carbon has 5 carbons.

   Therefore, the parent hydrocarbon is: $\text{pentanoic acid}$

3. Number the chain

   In carboxylic acids, numbering starts from the carboxyl carbon as carbon $1$.

   So the chain is numbered as: $C_1- C_2- C_3- C_4- C_5$

4. Locate substituents

   From the given structure:

   • There is a hydroxy group on carbon $3$
   • There is a methyl group on carbon $4$

5. Write the name in proper IUPAC format

   Substituents are written alphabetically:

   • hydroxy
   • methyl

   Hence the name is: $\boxed{\text{3-Hydroxy-4-methylpentanoic acid}}$

6. Check options

   • A: 3-Hydroxy-4-methylpentanoic acid ✅ Correct
   • B: 4,4-Dimethyl-3-hydroxybutanoic acid ❌ Wrong parent chain
   • C: 2-Methyl-3-hydroxypentan-5-oic acid ❌ Incorrect numbering/suffix usage
   • D: 4-Methyl-3-hydroxypentanoic acid ❌ Same locants, but substituents should be listed alphabetically, so hydroxy should come before methyl

Therefore, the correct answer is Option A.