To start, we can simplify the function by factoring out the common factor 4.
f(x)=4x^3-4x^2-16x+16=x^3-x^2-4x+4.
We could divide f(x) by (x-2) and solve the resulting quadratic equation.
We could, however, use another property of f(x).
We are given that.................................................... (x-2) is a factor
Knowing that the constant term is +4, and using the factor theorem, we can try (x+2) as a factor.
f(-2)=-8-4+8+4=0......................................................(x+2) is a factor
We note that f(x) has a sum of coefficients 1-1-4+4=0, equal to zero, this means that f(1)=0, or (x-1) is a factor...................................................(x-1)
Therefore all the factors of the function are (x+2)(x-2)(x-1)
and the roots are x={1,2,-2}