If $x_1,x_2$ are roots of $x^2+x+2=0and{x}_3,x_4$ are roots of $x^2+3x+4=0$ then the value of $\prod \left(x_i+x_j\right)fori,j\in \left\{1,2,3,4\right\}\&i<jis\lambda then\lambda is$ 

The roots are $x_1=\frac{-1+i\sqrt{7}}{2},x_2=\frac{-1-i\sqrt{7}}{2},x_3=\frac{-3+i\sqrt{7}}{2},x_4=\frac{-3-i\sqrt{7}}{2}$

Consider the expression

$\left(x_1+x_2\right)\left(x_1+x_3\right)\left(x_1+x_4\right)\left(x_2+x_3\right)\left(x_2+x_4\right)\left(x_3+x_4\right)$

By subsituting the above values,  we get the expression value as $132$