Answer
$f(x)=x^4+5x^2+4$
Work Step by Step
If $i$ is a zero of $f$ then the complex conjugate $-i$ is also a zero.
If $2i$ is a zero of $f$ then the complex conjugate $-2i$ is also a zero.
We have four zeros. We can find a four-degree polynomial.
$f(x)=a[(x-i)[x-(-i)](x-2i)[x-(-2i)]$
$f(x)=a(x^2-i^2)[x^2-(2i)^2]$
$f(x)=a(x^2+1)(x^2+4)$
$f(x)=a(x^4+4x^2+x^2+4)$
$f(x)=a(x^4+5x^2+4)$
$a=1$
$f(x)=1(x^4+5x^2+4)$
$f(x)=x^4+5x^2+4$
