Complex zeroes always come in pairs so another zero is -2i, which, together with 2i, creates the quadratic:
(x+2i)(x-2i)=x2+4.
Similarly, the irrational zero also has a counterpart -3-√5.
The quadratic is (x+3+√5)(x+3-√5)=x2+6x+9-5=x2+6x+4.
The quartic is;
f(x)=(x2+4)(x2+6x+4)=x4+6x3+4x2+4x2+24x+16=x4+6x3+8x2+24x+16.