The number of integral values of m for which the quadratic...

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

$⇒ m > \dfrac{- 1}{2}$ and

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

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

$⇒ {m}^{2} - 6m - 3 < 0$

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

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

$∴ m = 0, 1, 2, 3, 4, 5, 6$

$∴$ Number of integral values $= 7$