I don't have experience with solving these types of equations.
After looking around; I learned that I should try substituting a + bi in for a complex number in these types of equations.
$$w^2 + 3w^* + 2 = 0$$
I found that the real part is $a^2 - b^2 + 3a + 2$
I also found the imaginary part to be $2ab - 3b$
I'm not sure how to proceed.
Note, w* is the complex conjugate.