Let ${\mathbf{Z}}_1,{\mathbf{Z}}_2,{\mathbf{Z}}_3,{\mathbf{Z}}_4\in \mathrm{C}$ (none of z_i are zero and no two of z_i are equal for i = 1, 2, 3, 4) be such that 
$\begin{array}{r}{z}_1\left(z_1-z_2\right)\left(z_1-z_3\right)\left(z_1-z_4\right)=4\\ z_2\left(z_2-z_1\right)\left(z_2-z_3\right)\left(z_2-z_4\right)=4\\ z_3\left(z_3-z_1\right)\left(z_3-z_2\right)\left(z_3-z_4\right)=4\\ z_4\left(z_4-z_1\right)\left(z_4-z_2\right)\left(z_4-z_3\right)=4\end{array}$
Then which is/are CORRECT ?

Hence none of z_i are zero and no two of z_i are equal i
$\begin{array}{l}=1,2,3,4\\ \text{ Let }f\left(x\right)=\left(x-z_1\right)\left(x-z_2\right)\left(x-z_3\right)\left(x-z_4\right)\\ f^{\mathrm{\prime }}\left(x\right)=\left(x-z_2\right)\left(x-z_3\right)\left(x-z_4\right)+\left(x-z_1\right)\left(x-z_3\right)\left(x-z_4\right)\\ +\left(x-z_1\right)\left(x-z_2\right)\left(x-z_4\right)+\left(x-z_1\right)\left(x-z_2\right)\left(x-z_3\right)\\ f^{\mathrm{\prime }}\left(z_1\right)=\left(z_1-z_2\right)\left(z_1-z_3\right)\left(z_1-z_4\right)=\frac{4}{z_1}\end{array}$
Similarly,  $f^{\mathrm{\prime }}\left(z_2\right)=\frac{4}{z_2}$
$\begin{array}{l}{\mathrm{f}}^{\mathrm{l}}\left({\mathrm{z}}_3\right)=\frac{4}{{\mathrm{z}}_3},{\mathrm{f}}^{\mathrm{I}}\left({\mathrm{z}}_4\right)=\frac{4}{{\mathrm{z}}_4}\\ \Rightarrow {\mathrm{xf}}^{\mathrm{l}}\left(\mathrm{x}\right)=4\text{ has roots }{\mathrm{z}}_1{\mathrm{z}}_2{\mathrm{z}}_3\text{ and }{\mathrm{z}}_4\end{array}$
$\Rightarrow x{f}^{\mathrm{\prime }}\left(x\right)-4=0\text{ is a }4$ degree polynomial has leading coefficient is 4
$\begin{array}{l}\Rightarrow {\mathrm{xf}}^{\mathrm{\prime }}\left(\mathrm{x}\right)-4=4\mathrm{f}\left(\mathrm{x}\right)\\ \Rightarrow \frac{{\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)+1}=\frac{4}{\mathrm{x}}\end{array}$
$\begin{array}{l}\log \left(f\right(x)+1)=4\log x+a,a\text{ be any constant }\\ \log \left(f\right(x)+1)=\log \left(a{x}^4\right)\\ f\left(x\right)+1=a{x}^4\\ f\left(x\right)=a{x}^4-1\text{ comparing coefficient of }{x}^4\text{ we get }a=1\\ f\left(x\right)=x^4-1\end{array}$
then the roots are 1, –1, i, –i