If x + 1/x = 2, evaluate the rational expression (2x^2 + 2) / (3x^2 + 5x + 3). Provide the exact simplified fraction.

Introduction / Context:
This algebra simplification hinges on interpreting the condition x + 1/x = 2. For real numbers, this equality forces x = 1 (since x + 1/x ≥ 2 with equality at x = 1). Substituting x = 1 into the given rational expression yields an exact fraction.

Given Data / Assumptions:

• Condition: x + 1/x = 2 (real x).
• Expression to evaluate: (2x^2 + 2) / (3x^2 + 5x + 3).
• Principal arithmetic and exact fractions are expected.

Concept / Approach:
From the AM ≥ GM inequality or by direct algebra, x + 1/x = 2 implies x = 1. Plug this into the rational expression and simplify carefully to obtain the final fraction.

Step-by-Step Solution:

Verification / Alternative check:
Try any x ≠ 1 and the condition fails; thus no other real x satisfies the premise. Therefore the direct substitution is valid and unique.

Why Other Options Are Wrong:
1/2, 13/4, 7, and 2/11 are distractors that result from mis-substitution or arithmetic slips; none match the exact evaluation at x = 1.

Common Pitfalls:
Attempting to manipulate x symbolically without first using the key implication x = 1, or miscomputing the denominator 3 + 5 + 3.

Final Answer:
4/11