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

bx² + cx + a = 0
Roots are imaginary ⇒ c² − 4ab < 0⇒ c² < 4ab⇒ −c² > −4ab
3b² x² + 6bcx + 2c²
Since 3b² > 0
Given expression has a minimum value
Minimum value $=\frac{4\left(3{\mathrm{b}}^2\right)\left(2{\mathrm{c}}^2\right)-36{\mathrm{b}}^2{\mathrm{c}}^2}{4\left(3{\mathrm{b}}^2\right)}=-\frac{12{\mathrm{b}}^2{\mathrm{c}}^2}{12{\mathrm{b}}^2}=-{\mathrm{c}}^2>-4\mathrm{ab}$