Use the identity (a + b)^2 + (a − b)^2 = 2(a^2 + b^2) to evaluate: [ (238 + 131)^2 + (238 − 131)^2 ] / ( 238*238 + 131*131 ).

Introduction / Context:This problem is a direct application of the identity (a + b)^2 + (a − b)^2 = 2(a^2 + b^2). Recognizing this pattern quickly reduces the expression to a constant, independent of the specific values of a and b, as long as the structure matches the identity.

Given Data / Assumptions:

• a = 238, b = 131.
• Expression: [ (a + b)^2 + (a − b)^2 ] / ( a^2 + b^2 ).

Concept / Approach:Apply the identity to the numerator. Since the denominator equals a^2 + b^2, the ratio simplifies immediately. No numeric expansion is required, though you may compute to verify.

Step-by-Step Solution:(a + b)^2 + (a − b)^2 = 2(a^2 + b^2).Therefore the whole ratio = [2(a^2 + b^2)] / (a^2 + b^2) = 2.

Verification / Alternative check:Compute a^2 + b^2 numerically if desired and observe exact cancellation when doubling and dividing.

Why Other Options Are Wrong:

• 4, 8, 9, 1: These values appear if you square and add incorrectly or forget to divide by a^2 + b^2.

Common Pitfalls:Over-expanding and making arithmetic mistakes; ignoring the elegant identity that saves time.

Final Answer:2