[Math] Complex Conjugate roots with non real coefficients

Question

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist)

Is it possible for a polynomial with non-real coefficients to have complex conjugate roots?
If yes, could you give me an example of a quadratic equation with non-real coefficients that give complex conjugate solutions (except for the trivial cases such as I(x^2-4x+13)=0)

Thanks

Answer 1

Given a complex number z, a polynomial of degree two having $z$ and $\bar z$ as roots is $(x-z)(x-\bar z) = x^2 -(z + \bar z)x +z\bar z$.

$z +\bar z$ is real and $z\bar z$ is real too.

Thus to obtain a polynomial of degree two with conjugate complex roots and complex coefficients you only have trivial examples like $3i(x^2+1)$ or the one you mentioned above.