The image/structure of hydrocarbon $(X)$ is not provided in the prompt, so the exact parent chain cannot be independently reconstructed from the text alone. However, we can still analyze the given options using IUPAC naming rules and check which option is valid.

Step 1: Check whether each option follows IUPAC rules

Option B: $2$-Ethyl-$2,6$-diethylheptane

This is not a proper IUPAC name because:

• it repeats the substituent ethyl separately as both "ethyl" and "diethyl"
• all identical substituents should be combined

So this is invalid.

Option D: $2$-Ethyl-$3,6$-dimethylheptane

A name with parent chain as heptane and an ethyl substituent often suggests that a longer carbon chain may be possible. In such cases, IUPAC requires choosing the longest continuous chain. Very often such structures are better named as substituted octane rather than ethyl-substituted heptane.

So this is likely not the preferred IUPAC name.

Compare Options A and C

• A: $2,5,6$-Trimethyloctane
• C: $3,4,7$-Trimethyloctane

These describe the same kind of carbon skeleton named from opposite ends. By the lowest set of locants rule, we compare:

At the first point of difference:

• $2 < 3$

Hence, $2,5,6$ is preferred over $3,4,7$.

Therefore, between A and C, A is correct.

Step 2: Final conclusion

The valid and preferred IUPAC name is: