Let x = 1 + √2 + √3. Evaluate the expression: x^2 − 2x − 4 Choose the exact simplified value.

Introduction / Context: This problem checks algebraic simplification with surds (square roots). The best approach is to expand x^2 carefully, then simplify and cancel terms instead of approximating decimals.

Given Data / Assumptions:

• x = 1 + √2 + √3 • Required: x^2 − 2x − 4 • Keep the result in exact surd form

Concept / Approach: Compute x^2 using (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc, where: a = 1, b = √2, c = √3. Then substitute into x^2 − 2x − 4 and simplify systematically.

Step-by-Step Solution: 1) Let a = 1, b = √2, c = √3 2) Compute squares: a^2 = 1, b^2 = 2, c^2 = 3 3) Compute cross terms: 2ab = 2*1*√2 = 2√2 2ac = 2*1*√3 = 2√3 2bc = 2*√2*√3 = 2√6 4) So x^2 = 1 + 2 + 3 + 2√2 + 2√3 + 2√6 x^2 = 6 + 2√2 + 2√3 + 2√6 5) Compute 2x: 2x = 2 + 2√2 + 2√3 6) Now evaluate x^2 − 2x − 4: (6 + 2√2 + 2√3 + 2√6) − (2 + 2√2 + 2√3) − 4 7) Combine constants: 6 − 2 − 4 = 0 8) Cancel surd terms: 2√2 − 2√2 = 0 and 2√3 − 2√3 = 0 9) Remaining term: 2√6

Verification / Alternative check: A quick check: x is about 1 + 1.414 + 1.732 ≈ 4.146. Then x^2 ≈ 17.19. Compute x^2 − 2x − 4 ≈ 17.19 − 8.29 − 4 ≈ 4.90, and 2√6 ≈ 4.898, matching closely.

Why Other Options Are Wrong: • √6 is too small by a factor of 2. • 2√3, 3√3, 3√2 do not appear after cancellations and have different approximate sizes.

Common Pitfalls: • Forgetting the 2bc term = 2√6. • Making sign mistakes when subtracting 2x and 4.

Final Answer: 2√6