Congratulations to Fok Chi Kwong from Yuen Long Merchants Association Secondary School, Hong Kong on this solution.

We may find the required polynomial by starting from the expression :

x = 1 + \sqrt{2} + \sqrt{3}

. Squaring both sides and simplifying, we get

x - 1 = \sqrt{2} + \sqrt{3}

x^{2} - 2 x + 1 = 5 + 2 \sqrt{6}

x^{2} - 2 x - 4 = 2 \sqrt{6}

(x^{2} - 2 x - 4)^{2} = 24

x^{4} - 4 x^{3} + 4 x^{2} - 8 x^{2} + 16 x + 16 = 24

x^{4} - 4 x^{3} - 4 x^{2} + 16 x - 8 = 0

Thus p(x) = x⁴ - 4x³ - 4x² + 16x - 8 is the required polynomial.

Tony Cardell, State College Area High School, PA, USA, also sent in a good solution.