1. Write the structure of the substrate

   The compound is 3-bromo-2,2-dimethylbutane.

   Start from butane: $CH_3-CH_2-CH_2-CH_3$

   Put two methyl groups at C-2 and bromine at C-3:

   $CH_3-C(CH_3)_2-CH(Br)-CH_3$

2. Nature of reagent/solvent

   The reaction is carried out in ethanol $\left(C_2H_5OH\right)$.

   In alcoholic medium, alkyl halides can undergo solvolysis/substitution. Here, since the carbon bearing Br is adjacent to a quaternary-like crowded center, ionization can be followed by rearrangement to form a more stable carbocation.

3. Formation of carbocation

   Loss of $Br^-$ from

   $CH_3-C(CH_3)_2-CH(Br)-CH_3$

   gives a secondary carbocation at C-3:

   $CH_3-C(CH_3)_2-CH^+-CH_3$

4. Rearrangement of carbocation

   The adjacent carbon (C-2) is highly substituted. A 1,2-methyl shift from C-2 to the carbocation center (C-3) occurs, generating a more stable tertiary carbocation.

   After methyl shift, the positive charge moves to C-2:

   $CH_3-C^+(CH_3)-CH(CH_3)-CH_3$

   This is a tertiary carbocation.

5. Attack by ethanol

   Ethanol attacks the tertiary carbocation at C-2, followed by deprotonation, giving the ether:

   $CH_3-C(OC_2H_5)(CH_3)-CH(CH_3)-CH_3$

6. IUPAC name of the product

   Longest chain = butane.

   Substituents:

   • at C-2: ethoxy and methyl
   • at C-3: methyl

   Hence the product is:

   $\boxed{\text{2-Ethoxy-2,3-dimethylbutane}}$

7. Check options

   • A: 2-Ethoxy-3,3-dimethyl butane $\rightarrow$ incorrect structure
   • B: 1-Ethoxy-3,3-dimethyl butane $\rightarrow$ incorrect
   • C: 2-Ethoxy-2,3-dimethyl butane $\rightarrow$ correct
   • D: 2-Hydroxy-3,3-dimethyl butane $\rightarrow$ alcohol not expected with ethanol as nucleophile

Therefore, the major product is Option C.