Answer
See the detailed answer below.
Work Step by Step
Let's assume that at $t=0$, frame $S'$ overlaps with frame $S'$.
And at $t_1$, $S'$ is at $x=5$ m, and at $t_1$, $S'$ is at $x=25$ m, as shown in the figure below.
Now we need to use the Galilean transformations of position to find $x_1'$ and $x_2'$;
$$x=x'+vt$$
So, the first explosion at $t_1$ occurs at the position of
$$x'_1=x_1-vt_1$$
Plug the known;
$$x'_1=10-(5)(1)=\color{red}{\bf 5}\;\rm m$$
And the second explosion at $t_2$ occurs at the position of
$$x'_2=x_2-vt_2$$
Plug the known;
$$x'_2=20-(5)(5)=\color{red}{\bf -5}\;\rm m$$
