Finding remaining polynomial after finding complex factors

I want to express this polynomial as a product of linear factors:

$x^5 + x^3 + 8x^2 + 8$

I noticed that $\pm$i were roots just looking at it, so two factors must be $(x- i)$ and $(x + i)$, but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just do use long division but it turns out a little messy in this instance (for me at least) and was wondering if there was a simpler method of finding the remaining polynomial.

Apologies for the basic question!

Answer

If you divide

$$x^5 + x^3 + 8x^2 + 8$$ by $$(x-i)(x+i) = x^2+1$$ you will get $$x^3+8$$ which factors as $$(x^3+8) = (x+2)(x^2-2x+4)$$ which has a solution of $x=-2$

Now use quadratic formula to solve $x^2-2x+4=0$ to find other roots and factor if you wish.