Given, $\left|\begin{array}{cc}x^3+2 x^2+3 x-2 & x^2+2 x+4 \\ x^3-x^2-2 x-1 & 3 x^3-x^2+4 x-2\end{array}\right|$
$=a x^6+b x^5+c x^4+d x^3+e x^2+f x+g$
On expanding the determinant,
$\begin{aligned}\left(x^3+2 x^2+3 x-2\right) & \left(3 x^3-2 x^2+4 x-2\right) \\ & -\left(x^2+2 x+4\right)\left(x^3-x^2-2 x-1\right)\end{aligned}$
$\begin{aligned} \Rightarrow 3 x^6-2 x^5 & +4 x^4-2 x^3+6 x^5-4 x^4+8 x^3 \\ & -4 x^2+9 x^4-6 x^3+12 x^2 \\ & -6 x-6 x^3+4 x^2-8 x+4 \\ & -\left\{x^5-x^4-2 x^3-x^2+2 x^4\right. \\ & \left.-2 x^3-4 x^2-2 x+4 x^3-4 x^2-8 x-4\right\}\end{aligned}$
$\begin{aligned} \Rightarrow 3 x^6+4 x^5 & +9 x^4-6 x^3+12 x^2 \\ & -14 x+4-x^5-x^4+9 x^2+10 x+4\end{aligned}$
$\Rightarrow 3 x^6+3 x^5+8 x^4-6 x^3+21 x^2-4 x+8$
$=a x^6+b x^5+c x^4+d x^3+e x^2+f x+g$
$\Rightarrow \quad a=3, b=3, c=8, d=-6, e=21$,
$f=-4, g=8$
$\therefore a+b+c+d+e+f=25$