Evaluate a cubic identity under a linear condition: If a + b + 1 = 0, compute the value of a^3 + b^3 + 1 − 3ab.

Introduction / Context:
This is a direct application of the identity a^3 + b^3 + c^3 − 3abc = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Recognizing c = 1 and that a + b + 1 is given simplifies the expression to zero immediately.

Given Data / Assumptions:

• a + b + 1 = 0
• Target: a^3 + b^3 + 1 − 3ab

Concept / Approach:
Set c = 1 in the well-known identity. Because the factor (a + b + c) appears, if a + b + 1 equals zero, then the whole product must vanish, provided everything is finite. This makes the computation simple and robust.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 1, 3, −1: These contradict the factorization result which forces the value to zero under the given condition.

Common Pitfalls:

• Applying a^3 + b^3 formula incorrectly instead of the three-term identity.
• Forgetting to set c = 1 explicitly.

Final Answer: