First off, if x=-8 is a root, then it came from x+8 = 0 which makes (x+8) a factor of your polynomial.
Secondly, complex roots always come in conjugate pairs, so if -i is a root, then so is i and likewise if -5+i is a root then
-5-i is also a root.
So if x=-i is an answer, then x+i = 0 and (x+i) is a factor.
if x=i is an answer, then x-i=0 and (x-i) is a factor.
if x=-5+i is a root, then x+5-i =0 and (x+5-i) is a factor, and
if x=-5-i is a root, then x+5+i =0 and (x+5+i) is a factor.
So just multiply these all out.
(x+8)(x-i)(x+i)(x+5+i)(x+5-i) = 0
multiplying the conjugates makes life a little easier rather than multiplying in order. So,
(x+8)( x2+1)(x2+10x+26) =0
(x+8)(x4+10x3+27x2+10x+26)=0 and lastly,
x5 + 18x4+107x3+226x2+106x+208=0
Hope this helped.