Let $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+\mathrm{bx}+\mathrm{c}$ where $\mathrm{b},\mathrm{c}\in \mathrm{R}$. If f(x) is a factor of both ${\mathrm{x}}^4+6{\mathrm{x}}^2+25\text{ and }3{\mathrm{x}}^4+4{\mathrm{x}}^2+28\mathrm{x}+5$, then the least value of f(x) is

$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+\mathrm{bx}+\mathrm{c}$

f(x) is factor of both ${\mathrm{x}}^4+6{\mathrm{x}}^2+25\text{ and }3{\mathrm{x}}^4+4{\mathrm{x}}^2+28\mathrm{x}+5$

f(x) will also be factor of

$\begin{array}{l}3\left({\mathrm{x}}^4+6{\mathrm{x}}^2+25\right)-\left(3{\mathrm{x}}^4+4{\mathrm{x}}^2+28\mathrm{x}+5\right)=14\left({\mathrm{x}}^2-2\mathrm{x}+5\right)\\ \therefore f\left(x\right)={\mathrm{x}}^2-2\mathrm{x}+5 ={\left(x-1\right)}^2+4\ge 4\end{array}$