Smart way to solve (complex) polynomial equation

Question

Solve for all solutions: $z^4+z^3-3z^2-17z-30=0, z \in \mathbb{C} $, Premise: $-1+2i$ is a solution

I would factor $-1+2i$ out, guess another root, which can be factored out again, to get an polynomial of deg 2.

Is there another (smarter) way to get those solutions ?

Answer 1

Hint: $$-1-2i$$ is also a solution of your equation.

Answer 2

No, you dont guess another root. Since $-1+2i$ is a root and since the coefficients are real, then you know that $-1-2i$ is another root.