Recover and solve the intended equation (repair applied): Interpret the corrupted stem as ( (x + 4) / (x − 4) ) + ( (x − 4) / (x + 4) ) = 10 / 3. Find all real roots x.

Introduction / Context:
The original database text was garbled. Using the Recovery-First Policy, we interpret it as a classic rational-expression identity: (x + 4)/(x − 4) + (x − 4)/(x + 4) = 10/3. This kind of expression simplifies via a common denominator, producing an equation only in x^2. We then solve for x and check domain restrictions (x ≠ ±4).

Given Data / Assumptions:

• (x + 4)/(x − 4) + (x − 4)/(x + 4) = 10/3.
• Domain: x ≠ 4, x ≠ −4 (to avoid zero denominators).

Concept / Approach:
Use the identity (A/B) + (B/A) = (A^2 + B^2)/(AB). Here, A = x + 4 and B = x − 4. This yields a rational equation in x that simplifies to a quadratic in x^2. Solve and select the real roots allowed by the domain.

Step-by-Step Solution:

Verification / Alternative check:
Denominator check: x ≠ ±4, so ±8 are valid. Substitute x = 8 (or −8) to confirm the left-hand side equals 10/3 numerically.

Why Other Options Are Wrong:
±4 are excluded (division by zero). ±6 and 2 ± √3 do not satisfy the equation when substituted. “No real root” is false since ±8 work.

Common Pitfalls:
Dropping the domain restriction, or algebra errors when simplifying the rational sum can lead to spurious answers.

Final Answer:
± 8