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 + Ö2 + Ö3. Squaring both sides and simplifying, we get
x - 1 = Ö2+ Ö3

x2 - 2x + 1 = 5 + 2Ö6

x2 - 2x - 4 = 2Ö6

(x2 - 2x - 4)2 = 24

x4 - 4x3 + 4x2 - 8x2 + 16x + 16 = 24

x4 - 4x3 - 4x2 + 16x - 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.