If  x^3+ax+b is divisible by x^2+2x-2, then 

    x^3+ax+b = (x^2+2x-2)*(x+c),

where "c" is an unknown number. Now open the parentheses on the right. You will get

    x^3+ax+b = x^3 + 2x^2 - 2x + cx^2 + 2cx - 2c,  or

    x^3+ax+b = x^3 + (2+c)x^2 + (-2+2c)x - 2c.

Next, comparing the coefficients at x^2, x and the constant terms on both sides, you have these equalities

0 = 2 + c,       (1)       (coefficients at x^2)

a = -2 + 2c,     (2)       (coefficients at x)

b = -2c.         (3)       (constant terms).

Now from (1) you have c = -2;  from (2) you have a = -2+2*(-2) = -6;  and from (3) you have b = -2*(-2) = 4.

The summary is:  in order for x^3+ax+b be divisible by x^2+2x-2,

                 the following requirements must be in place: a = -6 and b = 4.  (They are necessary and sufficient conditions). 

                 Then  c = -2, and  indeed x^3 -6x + 4 is divisible by x^2+2x-2.