If $S=\{m\in R;x^2-2\left(1+3m\right)x+7\left(3+2m\right)=0$ by different roots}, then the number of elements in S is

$x^2-2\left(1+3m\right)x+7\left(3+2m\right)=0$

$a>0$ $\&D=b^2-4ac>0$

${\left(-2\left(1+3m\right)\right)}^2-4\times 7\left(3+2m\right)>0$

$9m^2-8m-20>0$

$\left(m-2\right)\left(m+\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$9$}\right.\right)>0$

$m\in \left(-\infty ,\frac{-10}{9}\right)\cup [2,\infty )$

Integral values of m are infinite