The number of integral values of m for which the quadratic expression 1 + 2 m   x 2 - 2 1 + 3 m x + 4 1…

$\because$ Quadratic expression is positive, hence $1+2m>0$ and $D<0$

$\Rightarrow m>\frac{-1}{2}$ and

$4{\left(1+3m\right)}^2-16 \left(1+m\right)\left(1+2m\right)<0$

$\Rightarrow 9m^2+1+6m-4\left(2m^2+3m+1\right)<0$

$\Rightarrow m^2-6m-3<0$

$\Rightarrow \left(m-3+2\sqrt{3}\right)\left(m-3-2\sqrt{3}\right)<0$

$\Rightarrow m\in \left(3-2\sqrt{3}, 3+2\sqrt{3}\right)$

$\therefore m=0, 1, 2, 3, 4, 5, 6$

$\therefore$ Number of integral values $=7$