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Written solution:

We can express this expression in base $10$ using the definition of bases: $$2b^3 + 2b + 1 - 2b^2 - 2b - 1$$$$= 2b^3 - 2b^2 = 2b^2(b - 1).$$

For this to be divisible by $3,$ either $b$ or $b - 1$ must be divisible by $3.$

The only answer choice that satisfies neither of these conditions is $8.$

Thus, E is the correct answer.