Good!

I guess this requires a bit of additional knowledge about polynomial equations with real coefficients and complex roots.

A polynomial that has both z1 and z2 as its roots, must be of the form:

(z-z1)(z-z2)(...) = 0

If such an equation has only real coefficients, that implies that not only z1 must be a root, but also its conjugate z1*.

(I suspect that is somewhere in your notes.)

In other words, your equation must be of the form:

$\displaystyle (z-z_1)(z-z_1^*)(z-z_2)(z-z_2^*) = 0$

If you expand this, which coefficients do you get?