If (a + b) : (a − b) = 5 : 3, find (a^2 + b^2) : (a^2 − b^2) in simplest form.

Introduction / Context:
Using the given ratio of (a + b) to (a − b), we can extract values for a and b up to a common factor, then compute the requested ratio for the squares.

Given Data / Assumptions:
(a + b) : (a − b) = 5 : 3 with real numbers allowing a > b for positivity.

Concept / Approach:
Let a + b = 5k and a − b = 3k. Solve for a and b, then compute a^2 + b^2 and a^2 − b^2 = (a − b)(a + b).

Step-by-Step Solution:
Add: (a + b) + (a − b) = 5k + 3k ⇒ 2a = 8k ⇒ a = 4k. Then b = (a + b) − a = 5k − 4k = k. a^2 + b^2 = (4k)^2 + k^2 = 16k^2 + k^2 = 17k^2. a^2 − b^2 = (a − b)(a + b) = 3k * 5k = 15k^2. Ratio = 17 : 15.

Verification / Alternative check:
Choose k = 1 ⇒ a = 4, b = 1 ⇒ (a + b)/(a − b) = 5/3 and (a^2 + b^2)/(a^2 − b^2) = 17/15.

Why Other Options Are Wrong:
25 : 9, 4 : 1, and 16 : 1 derive from squaring the given ratio or misapplying identities.

Common Pitfalls:
Squaring 5 : 3 directly; remember we are not finding (a + b)^2 : (a − b)^2 but a^2 + b^2 : a^2 − b^2.

Final Answer:
17 : 15