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!