Let f(x) = ax^{4} +2x^{3} -3x^{2} +bx -4 and g(x) = x^{2} -4

We have g(x) = x^{2} − 4 = (x-4) (x+2)

Given g(x) is a factor of f(x).

(x-2) and (x+2) are factors of f(x)

From factor theorem,

If (x-2) and (x+2) are factor of f(x) then f(2) = 0 and f(-2) = 0 respectively