Answer
$f(x)=-2x^3+6x^2+12x-16$
Work Step by Step
Step 1. Given $x=-2,1, 4$ as zeros, we can write the polynomial as $f(x)=k(x+2)(x-1)(x-4)=k(x^3-3x^2-6x+8)$ where $k$ is an unknown constant.
Step 2. Use the condition $f(2)=16$, we have $k(2^3-3(2)^2-6(2)+8)=16$, thus $k=-2$
Step 3. Thus we have $f(x)=-2x^3+6x^2+12x-16$
