The given time signal is:
$$u(t) = -3\sigma(t+4) + 6\sigma(t) - 3\sigma(t-4)$$
$\sigma$ - unit step function
The same signal can be describes with the following mathematical relation between $u(t)$ and $u_1(t)$:
$$u(t) = u_1(t-t_0) - u_1(t+t_0)$$
Now I have to determine the signal $u_1(t)$.
First attempt
$$ \begin{align} u(t) &= u(t)\\ -3\sigma(t+4)+6\sigma(t)-3\sigma(t-4) &= u_1(t-t_0)-u_1(t+t_0)\\ \color{red}{-3\sigma(t+4)+3\sigma(t)}\color{blue}{-3\sigma(t-4)+3\sigma(t)} &= \color{red}{u_1(t-t_0)}\color{blue}{-u_1(t+t_0)} \end{align} $$
At this point I see two problems:
- Ignoring the parameters, the signs does not look correctly. Expecting sign change between blue and red on the left side.
- Not ignoring the parameters, could the signs be right and should I assume $t_0 = 4$?
Second attempt
Canceled because of a good answer.
You need to see that by writing
$$u(t)=u_1(t-t_0)-u_1(t+t_0)$$
you simply decompose the signal $u(t)$ in two equally shaped signals with different signs and different time shifts. By looking at the signal $u(t)$ you see immediately that you can construct it by using a rectangular signal with height $3$ and width $4$, once shifting it to the left with a negative sign, and adding another such signal shifted to the right. Since both shifts need to be the same (with different signs), the signal $u_1(t)$ must be centered around $t=0$, i.e. it must be symmetric. So $u_1(t)$ is given by
$$u_1(t)=3[\sigma(t+2)-\sigma(t-2)]$$
which is a rectangle with height $3$ in the interval $[-2,2]$.
With a shift $t_0=2$ you get
$$u(t)=3[\sigma(t)-\sigma(t-4)-\sigma(t+4)+\sigma(t)]=-3\sigma(t-4)+6\sigma(t)-3\sigma(t+4)$$
which is indeed equal to the given $u(t)$.