Recovery and completion (minimal repair): Assume the intended relation is a^2 + b^2 + c^2 = 2(a + b + c) − 3. Find the value of (2a − 3b + 4c).

Introduction / Context:
The provided stem appears garbled with minus signs in unusual places, making the original unsolvable or ambiguous. Following the Recovery-First Policy, we minimally repair the statement to a standard, well-known form that uniquely determines a, b, and c: a^2 + b^2 + c^2 = 2(a + b + c) − 3. This allows a clean completion and preserves the algebraic learning objective.

Given Data / Assumptions:

• Repaired relation: a^2 + b^2 + c^2 = 2(a + b + c) − 3.
• a, b, c are real numbers.
• Find the exact value of 2a − 3b + 4c.

Concept / Approach:
Complete the square in three variables: (a − 1)^2 + (b − 1)^2 + (c − 1)^2 = 0. From this, deduce the unique solution a = b = c = 1. Then evaluate the required linear combination.

Step-by-Step Solution:

Verification / Alternative check:
Plug a = b = c = 1 into the original (repaired) equation: LHS = 1 + 1 + 1 = 3; RHS = 2(3) − 3 = 3; identity holds. The linear expression also evaluates consistently to 3.

Why Other Options Are Wrong:
Values 1, 2, 5, and 7 would require different values of a, b, c that do not satisfy the repaired identity.

Common Pitfalls:
Not recognizing the completing-the-square structure, or attempting to solve under the unrepaired, ambiguous stem. The minimal repair is standard and yields a unique solution.

Final Answer:
3