The set of all the values of m for which the equation  $x^2-2mx+m^2-1=0$ has at least one root in the internal [-2,4] is 

 $x^2-(m-1+m+1)x+(m-1)(m+1)=0$
So roots are m -1 and m+1
$-2\le m-1\le 4\Rightarrow -1\le m\le \mathrm{5.....}\left(1\right)$
$-2\le m+1\le 4\Rightarrow -3\le m\le \mathrm{3.....}\left(2\right)$
union of  (1) & (2) , $m\in [-3,5]$