Let z be a complex number satisfying ${\left|z\right|}^3+2z^2+4\overline{z}-8=0$, where $\overline{z}$ 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.

List-I	List-II
(P) ${\left|z\right|}^2\text{is equal to}$	(1) $12$
(Q) ${|z-\overline{z}|}^2\text{is equal to}$	(2) $4$
(R) ${\left|z\right|}^2+{|z+\overline{z}|}^2\text{is equal to}$	(3) $8$
(S) ${|z+1|}^2\text{is equal to}$	(4) $10$
	(5) $7$

The correct option is (P) → (2), (Q) → (1), (R) → (3), (S) → (5).

Step 1: Finding the real part of $z$

We are given the complex equation:

${\left|z\right|}^3+2z^2+4\overline{z}-8=0$

Taking the complex conjugate of the entire equation, since ${\left|z\right|}^3$ and $8$ are real numbers, we get:

${\left|z\right|}^3+2{\overline{z}}^2+4z-8=0$

Subtracting the original equation from its conjugate equation gives:

$2\left({\overline{z}}^2-z^2\right)+4\left(z-\overline{z}\right)=0$

Factoring out $\left(\overline{z}-z\right)$:

$2\left(\overline{z}-z\right)\left(\overline{z}+z\right)-4\left(\overline{z}-z\right)=0$

$2\left(\overline{z}-z\right)\left[z+\overline{z}-2\right]=0$

Since the imaginary part of $z$ is non-zero, $z\ne \overline{z}$, which means $\overline{z}-z\ne 0$. Therefore:

$z+\overline{z}-2=0\Rightarrow z+\overline{z}=2$

Since $z+\overline{z}=2\text{Re}\left(z\right)$, we have $\text{Re}\left(z\right)=1$.

Step 2: Finding the modulus $\left|z\right|$ and imaginary part

Let $z=1+iy$ with $y\ne 0$. Then:

$z^2={\left(1+iy\right)}^2=1-y^2+2iy$

$\overline{z}=1-iy$

Substitute $z$, $z^2$, and $\overline{z}$ into the original equation:

${\left|z\right|}^3+2\left(1-y^2+2iy\right)+4\left(1-iy\right)-8=0$

Simplifying the terms:

${\left|z\right|}^3+2-2y^2+4iy+4-4iy-8=0$

${\left|z\right|}^3-2y^2-2=0$

Since ${\left|z\right|}^2=1+y^2$, we have $y^2={\left|z\right|}^2-1$. Substituting this gives:

${\left|z\right|}^3-2\left({\left|z\right|}^2-1\right)-2=0$

${\left|z\right|}^3-2{\left|z\right|}^2=0$

${\left|z\right|}^2\left(\left|z\right|-2\right)=0$

Since $\left|z\right|>0$, we obtain:

$\left|z\right|=2$

Therefore:

${\left|z\right|}^2=4$

Also, $y^2={\left|z\right|}^2-1=4-1=3$.

Step 3: Evaluation of List-I entries

(P) ${\left|z\right|}^2$:

${\left|z\right|}^2=4$

Thus, (P) → (2).

(Q) ${\left|z-\overline{z}\right|}^2$:

$z-\overline{z}=\left(1+iy\right)-\left(1-iy\right)=2iy$

${\left|z-\overline{z}\right|}^2={\left|2iy\right|}^2=4y^2=4\times 3=12$

Thus, (Q) → (1).

(R) ${\left|z\right|}^2+{\left|z+\overline{z}\right|}^2$:

$z+\overline{z}=2\Rightarrow {\left|z+\overline{z}\right|}^2=2^2=4$

${\left|z\right|}^2+{\left|z+\overline{z}\right|}^2=4+4=8$

Thus, (R) → (3).

(S) ${\left|z+1\right|}^2$:

$z+1=\left(1+iy\right)+1=2+iy$

${\left|z+1\right|}^2=2^2+y^2=4+3=7$

Thus, (S) → (5).

Combining all the matches:

(P) → (2), (Q) → (1), (R) → (3), (S) → (5)