Find the coefficients of this polynomial. Given $f(x)=x^4 +ax^3 +bx^2 -3x +5$ , $  a,b\in \mathbb{R}$ and knowing that $i$ is a root of this polynomial. what are $a,b$? 
My work until now: since all the coefficients are real, I now know that $i$ and $-i$ are roots of this polynomial, there must be two more roots. I tried to use vietta but it didn't really help, I get that $x_3+x_4=-a$ and $i(-i)x_3x_4=5$. What am I missing? 
Appreciate any help.

$$f(i)=i^4 + a i^3 + b i^2 - 3 i + 5=-i a-b+(6-3 i)=0$$
$$6-b-i(3+a)=0$$
as $a,b\in\mathbb{R}$ we have $6-b=0,3+s=0$
$a=-3,b=6$
