If the roots of $x^4-8x^3+23x^2+kx+12=0$  are such that the difference of the two roots is equal to the difference of other two then $k=$

$\alpha +\delta =\beta +\gamma =4$

$\alpha +\beta +\gamma +\delta =8$

$2\left(\alpha +\delta \right)=8$

$\alpha +\delta =4$

$x^4-8x^3+23x^2+kx+12=\left(x^2-x\left(\alpha +\delta \right)+\alpha \delta \right)\left(x^2-x\left(\beta +\gamma \right)+\beta \gamma \right)$

$=\left(x^2-4x+p_1\right)\left(x^2-4x+p_2\right)$

$=x^4-4x^3+p_2{x}^2-4x^3+16x^2-4p_{2}x+p_1{x}^2-4x{p}_1+p_1{p}_2$

Coefficient of $x^2\to 23=p_2+16+p_1$

$p_1+p_2=7$

Coefficients of $x\to k=-4p_2-4p_1$

$=-4\left(p_1+p_2\right)$

$=-4\left(7\right)$

$=-28$