Given x = 4ab/(a + b), simplify the expression: (x + 2a)/(x − 2a) + (x + 2b)/(x − 2b) Assume a and b are real numbers such that all denominators are non-zero. Choose the correct simplified value.

Introduction / Context:
This question tests algebraic simplification with rational expressions. The value of x is given as a symmetric expression in a and b, and the goal is to simplify two complicated fractions into a clean constant by careful substitution and cancellation.

Given Data / Assumptions:

• x = 4ab/(a + b) • Expression: (x + 2a)/(x − 2a) + (x + 2b)/(x − 2b) • Assume denominators are not zero (a + b ≠ 0, x ≠ 2a, x ≠ 2b)

Concept / Approach:
Substitute x in each fraction, combine terms over the common denominator (a + b), and simplify. Often, factors like (a + b) cancel. The structure is designed so that the final result becomes a constant independent of a and b (when defined).

Step-by-Step Solution:
1) Substitute x = 4ab/(a + b) into the first fraction: x − 2a = 4ab/(a + b) − 2a = [4ab − 2a(a + b)]/(a + b) = [4ab − 2a^2 − 2ab]/(a + b) = [2ab − 2a^2]/(a + b) = 2a(b − a)/(a + b) x + 2a = [4ab + 2a(a + b)]/(a + b) = [6ab + 2a^2]/(a + b) = 2a(a + 3b)/(a + b) 2) So (x + 2a)/(x − 2a) = [2a(a + 3b)]/[2a(b − a)] = −(a + 3b)/(a − b) 3) Similarly for the second fraction: (x + 2b)/(x − 2b) = (3a + b)/(a − b) 4) Add them: −(a + 3b)/(a − b) + (3a + b)/(a − b) = [−a − 3b + 3a + b]/(a − b) = (2a − 2b)/(a − b) = 2

Verification / Alternative check:
Choose a = 2, b = 1: x = 4*2*1/(3) = 8/3. Compute: (x + 4)/(x − 4) + (x + 2)/(x − 2) = (20/3)/(−4/3) + (14/3)/(2/3) = −5 + 7 = 2. Confirms.

Why Other Options Are Wrong:
• 0, 1, 4, −2: the algebra simplifies exactly to 2 (when defined), so these cannot match.

Common Pitfalls:
• Dropping the (a + b) denominator too early. • Sign mistakes when converting (b − a) into −(a − b).

Final Answer:
2