Given x + a/x = 1 (with x ≠ 0), evaluate the expression (x^2 + x + a) / (x^3 − x^2) in simplest terms.

Introduction / Context:
Algebraic constraints like x + a/x = 1 are powerful: when multiplied through, they produce a quadratic relation connecting x and a. This relation can simplify more complicated rational expressions by substitution and cancellation.

Given Data / Assumptions:

• x + a/x = 1, with x ≠ 0.
• Target: S = (x^2 + x + a) / (x^3 − x^2).
• All work is exact; simplify fully.

Concept / Approach:
From x + a/x = 1, multiply both sides by x to get a polynomial constraint x^2 − x + a = 0. Use this to replace a wherever it appears. Then factor common terms and cancel responsibly to obtain a compact result in terms of a alone.

Step-by-Step Solution:

Verification / Alternative check:
Pick a feasible pair satisfying x^2 − x + a = 0 (e.g., choose x, compute a), then evaluate S numerically and compare with −2/a; they match.

Why Other Options Are Wrong:
−2 and −a/2 treat a as if equal to x(x − 1) without inversion; 2/a misses the negative sign; 2/(1 − a) is unrelated to the derived identity.

Common Pitfalls:
Forgetting to multiply the initial constraint by x, canceling x terms incorrectly, or losing the negative when substituting x(x − 1) = −a.

Final Answer:
-2/a