If A = (x^8 - 1)/(x^4 + 1) and B = (y^4 - 1)/(y^2 + 1), with x = 2 and y = 9, then what is the value of A^2 + 2AB + AB^2?

Introduction / Context:
This problem tests your ability to handle algebraic expressions with exponents and to substitute numerical values carefully. It also reinforces the idea that sometimes a complicated looking expression can be simplified by recognising a pattern similar to a known algebraic identity, even if it is not exactly the same as the standard form used in textbooks.

Given Data / Assumptions:

• A = (x^8 - 1) / (x^4 + 1)
• B = (y^4 - 1) / (y^2 + 1)
• x = 2 and y = 9
• Required expression: A^2 + 2AB + AB^2
• All variables represent real numbers and denominators are non zero.

Concept / Approach:
The expression A^2 + 2AB + AB^2 looks similar to the expansion of a square, but the last term is AB^2 instead of B^2. Therefore we cannot directly use the standard identity (A + B)^2. The safe approach is to compute A and B numerically first and then substitute these values into the given expression. Care must be taken with powers such as x^8 and y^4 so that arithmetic errors are avoided.

Step-by-Step Solution:
Step 1: Compute x^4 for x = 2. Since 2^4 = 16, we have x^4 = 16. Step 2: Compute x^8 for x = 2. Since x^8 = (x^4)^2 = 16^2 = 256. Step 3: Find A using A = (x^8 - 1)/(x^4 + 1) = (256 - 1)/(16 + 1) = 255/17 = 15. Step 4: Compute y^2 for y = 9; this gives y^2 = 81. Step 5: Compute y^4 for y = 9; this gives y^4 = (y^2)^2 = 81^2 = 6561. Step 6: Find B using B = (y^4 - 1)/(y^2 + 1) = (6561 - 1)/(81 + 1) = 6560/82 = 80. Step 7: Substitute A = 15 and B = 80 into the expression A^2 + 2AB + AB^2. Step 8: Compute A^2 = 15^2 = 225. Step 9: Compute 2AB = 2 × 15 × 80 = 2400. Step 10: Compute AB^2 = 15 × 80^2 = 15 × 6400 = 96000. Step 11: Add the three parts: 225 + 2400 + 96000 = 98625.

Verification / Alternative check:
You can quickly check the calculations by using a calculator or by grouping terms: first add 2400 and 96000 to get 98400, then add 225 to get 98625. Since A and B are both positive integers here, the final value should also be a reasonably large positive number, which matches 98625 and rules out much smaller options.

Why Other Options Are Wrong:
96475, 92425 and 89125 are close in magnitude but result from small arithmetic mistakes such as squaring 80 incorrectly or adding the final terms wrongly. The value 9025 might come from mistakenly evaluating (A + B)^2 instead of A^2 + 2AB + AB^2, but that is not the given expression. None of these match the correct computed sum of 98625.

Common Pitfalls:
A common mistake is misreading the powers in x^8 or y^4 and computing only x^4 or y^2. Another error is to assume the expression equals (A + B)^2 and simplify it directly to A^2 + 2AB + B^2, which would give a completely different result. Always read the expression carefully and compute powers step by step to avoid such slips.

Final Answer:
Therefore, the value of A^2 + 2AB + AB^2 is 98625.