Complex Numbers and Quadratic Equations: JEE Advanced 2023 Mathematics question

Let $z$ be a complex number satisfying $|z|^{3}+2 z^{2}+4 \bar{z}-8=0$, where $\bar{z}$ denotes the complex conjugate of $z$. Let the imaginary part of $z$ be nonzero. Match each entry in List-I to the correct entries in List-II. \textbf{List-I} (P) $|z|^{2}$ is equal to (Q) $|Z-\bar{Z}|^{2}$ is equal to (R) $|Z|^{2}+|Z+\bar{Z}|^{2}$ is equal to (S) $|z+1|^{2}$ is equal to \textbf{List-II} (1) 12 (2) 4 (3) 8 (4) 10 (5) 7 The correct option is: