If the roots of the equation  $b{x}^2+cx+a=0$  be imaginary, then for all real values of  $x,$ the expression  $3b^2{x}^2+6bcx+2c^2$  is

Given equation is $c^2-4ab<0$ ….. ( 1 )

Given that the roots of  Eq. ( 1 )  are  Imaginary (non-real),

therefore, $\left(\because b^2-4ac<0\right)$        $\Rightarrow c^2<4ab$

$3b^2{x}^2+6bcx+2c^2.$

and the given expression is $=3b^2\left[x^2+2\frac{c}{b}x\right]+2c^2$

  $=3b^2\left[x^2+2\frac{c}{b}x+\frac{c^2}{b^2}-\frac{c^2}{b^2}\right]+2c^2$

        $\left(add and subtract\frac{c^2}{b^2}\right)$      

    $=3b^2{\left(x+\frac{c}{b}\right)}^2+2c^2-3b^2\left(\frac{c^2}{b^2}\right)$ 

   $=3b^2{\left(x+\frac{c}{b}\right)}^2-c^2$   

  $\ge -c^2>-4ab$  

        $\left(\because c^2<4ab\right)$