Higher degree polynomial with complex roots

I'm working on the following problem:

$$r^4 - 3r^2 -4r = 0$$

I factor out one $r$ and leaving me $r(r^3 - 3r -4) = 0$. One real root is $r=0$, and I'm unable to find the other ones. I tried using synthetic division but it didn't help. I tried googling synthetic division with complex root problems, but all the videos use examples that are given a complex solution in order to solve the other roots. So what could be a good approach in this problem?

Answer

Hint. Applying Cardano's formula (see the link above) to the reduced equation

$$r^3 - 3r -4=0,$$ one gets the real root

$$r_1=\left(2-\sqrt{3}\right)^{1/3}+\left(2+\sqrt{3}\right)^{1/3}$$

and the two complex roots

$$r_{2}^{\pm}=-\frac12 \left(2- \sqrt{3}\right)^{1/3} \left(1\pm i \sqrt{3}\right)-\frac{1}{2} \left(2+\sqrt{3}\right)^{1/3}\left(1\mp i \sqrt{3}\right) .$$