Answer
$f(x)=2x^4+12x^3+20x^2+12x+18$
See graph.
Work Step by Step
Step 1. Given a zero $x=i$, we can identify another zero $x=-i$
Step 2. With a zero $x=-3$ of multiplicity of 2, we have $(x+3)^2$ as a factor.
Step 3. For $n=4$, we can write the polynomial as
$f(x)=(x-i)(x+i)(x+3)^2=(x^2+1)(x^2+6x+9)=x^4+6x^3+10x^2+6x+9$
Step 4. Let $x=-1$; we have
$f(-1)=(-1)^4+6(-1)^3+10(-1)^2+6(-1)+9=1-6+10-6+9=8$
Thus, we are missing a factor of $2$ for the polynomial.
Step 5. We conclude the polynomial should be
$f(x)=2x^4+12x^3+20x^2+12x+18$
Step 6. See graph for the zeros and $f(-1)$
