bpt $\Leftrightarrow \sqrt{x+3}+x-2 \ge \sqrt{2x^2-6x+14}$ $(x \ge-3)$$\Leftrightarrow \begin{cases}\sqrt{x+3}+x \ge 2 \\ x+3+(x-2)^2+2(x-2)\sqrt{x+3} \ge2x^2-6x+14 \end{cases}$
$\Leftrightarrow \begin{cases}x \ge \dfrac{5-\sqrt{21}}{2} \\ 2(x-2)\sqrt{x+3} \ge x^2-3x+7 \end{cases}$
$\Leftrightarrow \begin{cases}x \ge 2 \\ 4(x-2)^2(x+3) \ge (x^2-3x+7)^2 \end{cases}$
$\Leftrightarrow \begin{cases}x \ge2 \\ -(x^2-5x+1)^2 \ge0 \end{cases}$
$\Leftrightarrow \begin{cases}x \ge 2 \\ x^2-5x+1=0 \end{cases}\Leftrightarrow \boxed{x=\frac{5+\sqrt{21}}{2}}$