# The number of integral values of m for which the equation $\left ( 1+m^{2} \right )x^{2}-2\left ( 1+3m \right )x+\left ( 1+8m \right )=0$ has no real root is :  Option 1) $1$ Option 2) $2$ Option 3) infinitely many Option 4) $3$

Given equation is

$\left ( 1+m^{2} \right )x^{2}-2\left ( 1+3m \right )x+\left ( 1+8m \right )=0$

given eq. has no real root

$\Rightarrow D<0$

$\Rightarrow4\left ( 1+3m \right )^{2}-4\left ( 1+m^{2} \right )\left ( 1+8m \right )<0$

$\Rightarrow 4(1+6m+9m^2)-4(1+8m+m^2+8m^3)<0$

$\Rightarrow -8m(2m-1)^{2}<0$

Infinite value of m satisfies this.

Option 1)

$1$

Option 2)

$2$

Option 3)

infinitely many

Option 4)

$3$

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