Determine a polynomial p(x) of degree 4, with zero points, 1,2,√2,-√2 for which p(-1) = 2

x = 1 means (x - 1) is a factor.
x = √2 means (x - √2) is a factor.
x = 2 means (x - 2) is a factor.
x = -√2 means (x + √2) is a factor.

a(x - 1)(x - 2)(x + √2)(x - √2) = 0
a(x^2 - 3x + 2)(x^2 - 2) = 0
a(x^4 - 3x^3 + 2x^2 - 2x^2 + 6x - 4) = 0
a(1 + 3 + 2 - 2 - 6 - 4) = 2
-6a = 2
a = -1/3

p(x) = (-1/3) (x^4 - 3x^3 + 6x - 4) = (-1/3)x^4 + x^3 - 2x + 4/3

a(x-r1)(x-r2)(x-r3)(x-r4).

r1, r2, r3 and r4 are the four roots.
The value of a is chosen to make p(-1) = 2. Plug in x = -1 after you plug in the four roots, and solve for a.

If the zero points are so, then your polynomail must be of the form :
a*(x-1)(x-2)(x-sqrt(2))(x+sqrt(2)).
Then you calculate it's value at -1 and adjust a so that it gives 2.