Let $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{ax}}^2-\mathrm{bx}+{\mathrm{c}}^2,\mathrm{b}\ne 0$ and $\mathrm{f}\left(\mathrm{x}\right)\ne 0$ for all $\mathrm{x}\in \mathrm{R}$. Then

Here, ${\mathrm{ax}}^2-\mathrm{bx}+{\mathrm{c}}^2=0$ does not have real roots. So,

$\mathrm{D}<0\Rightarrow {\mathrm{b}}^2-4{\mathrm{ac}}^2<0\Rightarrow \mathrm{a}>0$

Therefore, f(x) is always positive. So,

$\mathrm{f}\left(2\right)>0\Rightarrow 4\mathrm{a}-2\mathrm{b}+{\mathrm{c}}^2>0$