Let x < 1, then value of $\left|\begin{array}{ccc}{\mathrm{x}}^2+2& 2\mathrm{x}+1& 1\\ 2\mathrm{x}+1& \mathrm{x}+2& 1\\ 3& 3& 1\end{array}\right|$ is

Applying ${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2\text{ and }{\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_3$ reduce the determinant to

$\begin{array}{l}\left|\begin{array}{ccc}{x}^2-2x+1& x-1& 0\\ 2x-2& x-1& 0\\ 3& 3& 1\end{array}\right|=(x-1{)}^3-2(x-1{)}^2\\ =(x-1{)}^2(x-1-2)\\ =(x-1{)}^2(x-3),\end{array}$

which is clearly negative for x < 1.