Answer
$f(x)=-2x^4-2x^3+2x^2-2x+4$
Work Step by Step
If $i$ is a zero of $f$ then the complex conjugate $-i$ is also a zero.
$f(x)=a[(x-(-2)](x-1)(x-i)[x-(-i)]$
$f(x)=a(x+2)(x-1)(x-i)(x+i)$
$f(x)=a(x^2-x+2x-2)[x^2-i^2]$
$f(x)=a(x^2+x-2)(x^2+1)$
$f(x)=a(x^4+x^2+x^3+x-2x^2-2)=a(x^4+x^3-x^2+x-2)$
$f(0)=a(0^4+0^3-0^2+0-2)=-4$
$a(-2)=4$
$a=-2$
$f(x)=-2(x^4+x^3-x^2+x-2)$
$f(x)=-2x^4-2x^3+2x^2-2x+4$
