Simplify the algebraic expression b − [b − (a + b) − {b − (b − a − b)} + 2a] completely and express the result in terms of a and b only.

Introduction / Context:
This question checks a candidate ability to simplify a nested algebraic expression involving brackets and plus or minus signs. It is common in algebra and aptitude tests to see whether learners carefully handle signs and groupings. Correct simplification requires methodical expansion of brackets and combination of like terms in terms of a and b.

Given Data / Assumptions:
- Expression: b − [b − (a + b) − {b − (b − a − b)} + 2a].
- Variables a and b are real numbers.
- We must simplify completely and express the final result only using a and b.
- Standard rules for addition, subtraction, and distribution over brackets apply.

Concept / Approach:
The key idea is to simplify from the innermost brackets outward. First simplify (a + b) and (b − a − b), then deal with {b − (b − a − b)}, and then with the larger square bracket. Throughout the process, we combine like terms and keep track of signs carefully. Finally, we subtract the simplified bracket from b to obtain the final expression.

Step-by-Step Solution:
Start with the inner expression (b − a − b) = −a. Then {b − (b − a − b)} becomes b − (−a) = b + a. Now inside the square brackets we have: b − (a + b) − (b + a) + 2a. Compute b − (a + b) = b − a − b = −a. So the bracket simplifies to −a − (b + a) + 2a. This becomes −a − b − a + 2a = (−a − a + 2a) − b = 0 − b = −b. Original expression is b − [ −b ] = b + b = 2b.

Verification / Alternative check:
We can verify by substituting simple values, for example a = 1, b = 2. Compute the original expression numerically and we obtain 4. Compute 2b with b = 2 and we also get 4. Trying a different pair, such as a = 3 and b = 5, again gives matching values. This confirms that the simplified expression 2b is correct.

Why Other Options Are Wrong:
Option 1 suggests the expression simplifies to a constant, which is impossible because the variable b clearly remains in the result.
Option 0 would mean the whole expression cancels out, which is not consistent with the step by step algebraic simplification.
Option 3b would arise if one of the minus signs or bracket expansions were mishandled, adding an extra b term incorrectly.

Common Pitfalls:
A frequent error is distributing a minus sign incorrectly when removing brackets, especially with nested expressions. Another common mistake is forgetting to simplify the inner bracket b − a − b correctly. Careful, systematic work from the innermost part outward is the safest strategy.

Final Answer:
The simplified value of the expression is 2b.