The number of integral values of $\alpha$ for which the equation $\left|x-1\right|-2\left|x-3\right|+\left|x+2\right|={\alpha }^2-2\alpha +1$ has atleast one solution, is

$f\left(x\right)=\left|x-1\right|-2\left|x-3\right|+\left|x+2\right|$

Range of $f\left(x\right)$ is $[-7,7]$

$f\left(x\right)={\alpha }^2-2\alpha +1$will have a solution if

$$\begin{gathered} \Rightarrow {\alpha }^2-2\alpha +1\le 7 \\ \Rightarrow {(\alpha -1)}^2\le 7 -\sqrt{7}+1\le \alpha \le \sqrt{7}+1 \end{gathered}$$

number of integral values of $\alpha$ is $5$ Ans.