If the roots of x⁴-8x³+23x²+kx+12=0 are such that the difference of the two roots is equal to the difference of other two then k = _____

$$\begin{gathered} \begin{array}{l}\because \mathrm{\alpha }-\mathrm{\beta }=\mathrm{\gamma }-\mathrm{\delta }\\ \Rightarrow \mathrm{\alpha }+\mathrm{\delta }=\mathrm{\beta }+\mathrm{\gamma }=4\\ {\mathrm{S}}_1=\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }+\mathrm{\delta }=8\\ \Rightarrow 2(\mathrm{\alpha }+\mathrm{\delta })=8\Rightarrow \mathrm{\alpha }+\mathrm{\delta }=4\\ {\mathrm{x}}^4-8{\mathrm{x}}^3+23{\mathrm{x}}^2+\mathrm{kx}+12\end{array} \\ \begin{array}{l}={\left({\mathrm{x}}^2-\mathrm{x}(\mathrm{\alpha }+\mathrm{\delta })+\mathrm{\alpha \delta }\right)}^2\\ =\left({\mathrm{x}}^2-4\mathrm{x}+{\mathrm{P}}_1\right)\left({\mathrm{x}}^2-4\mathrm{x}+{\mathrm{P}}_2\right)\end{array} \end{gathered}$$
Coefficient of x² is 
${\mathrm{P}}_2+16+{\mathrm{P}}_1=23\Rightarrow {\mathrm{P}}_1+{\mathrm{P}}_2=7$
Coefficient of x is -4P₂-4P₁=K
$\begin{array}{l}\Rightarrow -4\left({\mathrm{P}}_1+{\mathrm{P}}_2\right)=\mathrm{K}\\ \Rightarrow \mathrm{K}=-4\left(7\right)=-28\end{array}$