$$\begin{gathered} Given \\ \sqrt{{\left(x+1\right)}^2+{\left(x+2-2\right)}^2}+\sqrt{{\left(x-2\right)}^2+{\left(x+2-5\right)}^2} \end{gathered}$$

$$\begin{gathered} P\left(x,x+2\right) and A\left(2,5\right), B\left(-1,2\right) \\ P lies on the line y =x+2 \end{gathered}$$

$$\begin{gathered} Since PA +PB is minimum and P lies on y =x +2 \\ And both A, B lies on the same side of the line y=x+2 \end{gathered}$$

$$\begin{gathered} The image of P with respect to the line y=x+2 \\ \frac{h-2}{1}=\frac{k-5}{-1}=\frac{2}{2};h-2=1;h=3 \end{gathered}$$

$$\begin{gathered} k-5=-1;k=4 \\ Hence the image is \left(3,4\right) \end{gathered}$$

$$\begin{gathered} A\text{'}B is the minimum value of PA+PB \\ Hence, A\text{'}B is \sqrt{16+4}=\sqrt{20} \end{gathered}$$

$$\begin{gathered} Here \alpha =20 \\ It is even number, It has two prime factors 2,5 \end{gathered}$$

$Sum of the digits of \alpha is 2.$