In algebra, let a and b be real numbers such that a - b = 2 and a^3 - b^3 = 26. Based on these conditions, what is the numerical value of the squared sum (a + b)^2?

Introduction / Context:
This aptitude question checks your command of algebraic identities and your ability to connect given information about a - b and a^3 - b^3 with the required expression (a + b)^2. Instead of solving for a and b directly, the goal is to manipulate identities efficiently, which is a common pattern in competitive exams.

Given Data / Assumptions:
- a and b are real numbers.
- a - b = 2.
- a^3 - b^3 = 26.
- We are required to find (a + b)^2.

Concept / Approach:
The key identity is a^3 - b^3 = (a - b)(a^2 + ab + b^2). From this we can get a^2 + ab + b^2. We then express a^2 + b^2 and ab in terms of (a - b)^2 and finally compute (a + b)^2 using the identity (a + b)^2 = a^2 + b^2 + 2ab.

Step-by-Step Solution:
From a^3 - b^3 = (a - b)(a^2 + ab + b^2), substitute the given values: 26 = 2 * (a^2 + ab + b^2). So a^2 + ab + b^2 = 13. Let S = a^2 + b^2 and P = ab. Then S + P = 13. Also, (a - b)^2 = a^2 + b^2 - 2ab = S - 2P = 4, since a - b = 2. We now have the system S + P = 13 and S - 2P = 4. Subtract the second equation from the first: (S + P) - (S - 2P) = 13 - 4 which gives 3P = 9, so P = 3. Substitute P = 3 into S + P = 13 to get S = 10. Now (a + b)^2 = S + 2P = 10 + 2 * 3 = 16.

Verification / Alternative check:
We can quickly verify by trying to reconstruct a and b. From S = a^2 + b^2 = 10 and P = ab = 3, we get (a + b)^2 = 16, so a + b = 4 or -4. Combined with a - b = 2, solving gives a = 3, b = 1 or a = -1, b = -3. In both cases a^3 - b^3 = 26 and (a + b)^2 = 16, confirming the result.

Why Other Options Are Wrong:
9, 4 and 12 arise if you misapply identities, confuse (a + b)^2 with a^2 + b^2, or incorrectly solve for S and P. 25 is just a random square and does not satisfy the given equations when checked back.

Common Pitfalls:
Many learners try to find a and b directly, which is longer and more error prone. Another common mistake is to forget that (a - b)^2 = a^2 + b^2 - 2ab, not a^2 - b^2. Mixing up coefficients when solving the simultaneous equations for S and P also leads to wrong answers.

Final Answer:
The required value of the squared sum is 16.