Let x = 1 + √2 + √3. Compute the exact value of the expression 2x^4 − 8x^3 − 5x^2 + 26x − 28.

Introduction / Context:
This problem evaluates a higher-degree polynomial at x = 1 + √2 + √3. Such expressions often simplify to a neat surd multiple (like √6), provided the coefficients are crafted to cancel many terms. The goal is to compute the exact value without resorting only to decimal approximations.

Given Data / Assumptions:

• x = 1 + √2 + √3
• Target expression: 2x^4 − 8x^3 − 5x^2 + 26x − 28
• We use algebraic expansion and rational-surd simplifications.

Concept / Approach:
Although direct expansion is possible, a structured approach helps: set t = √2 + √3 so x = 1 + t. Compute needed powers of t using (√2 + √3)^2 = 5 + 2√6 and recognize patterns where conjugate pairs or symmetric terms cancel. After obtaining x^2, x^3, and x^4, substitute into the polynomial and simplify surd parts.

Step-by-Step Solution:
Let t = √2 + √3 ⇒ t^2 = 5 + 2√6.Then x = 1 + t, so compute x^2, x^3, x^4 via binomial expansions.Substitute x^2, x^3, x^4 into 2x^4 − 8x^3 − 5x^2 + 26x − 28.Collect rational terms and surd terms (multiples of √6). Many terms cancel by design.The final simplified form evaluates exactly to 6√6.

Verification / Alternative check:
Numerically, √2 ≈ 1.4142, √3 ≈ 1.7321, so x ≈ 4.1463. Evaluating the polynomial gives approximately 14.6969. Since √6 ≈ 2.4495, 6√6 ≈ 14.6969, confirming the exact value matches.

Why Other Options Are Wrong:
0, 3√6, 2√6, and √6 are either too small or do not match the computed numerical value. Only 6√6 aligns with the precise simplification and the numerical check.

Common Pitfalls:
Mistakes occur when expanding powers of (√2 + √3) or forgetting that √2 * √3 = √6. Sign errors in the polynomial substitution also commonly lead to incorrect constants or surd coefficients.

Final Answer:
6√6