Let the sixth term in the binomial expansion of \(\left(\sqrt{2^{\log _2}\left(10-3^x\right)}+\sqrt[5]{2^{(x-2) \log _2 3}}\right)^m\), in the increasing powers of \(2^{(x-2) \log _2 3}\), be \(21\) . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an \(A.P.\), then the sum of the squares of all possible values of \(x\) is \(.........\).

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.