If $p,q,r\in \mathbb{C}$, how would one describe the curve $\mathrm{Re}\left(pz^2+qz+r\right)=0.$
If I write $p=p_1+ip_2$, $q=q_1+iq_2$, $r=r_1+ir_2$, and $z=x+iy$, then
$\mathrm{Re}\left(pz^2+qz+r\right)=p_1\left(x^2-y^2\right)-2p_2xy+q_1x-q_2y+r_1=0.$
Can I therefore say that the above is a polynomial in $x$ and $y$? If it is right, would that be enough? Are there any cases, I must consider?