How are these two discrete time signals equal?
They are indeed equal. The first term is

$$-3(-2)^nu[n]=(-3)(-2)(-2)^{n-1}u[n]=6(-2)^{n-1}u[n]\tag{1}$$

The value of $(1)$ at $n=0$ is $6/(-2)=-3$, so we can rewrite it as

$$-3\delta[n]+6(-2)^{n-1}u[n-1]\tag{2}$$

Adding the second term $(-2)^{n-1}u[n-1]$ results in

$$x[n]=-3\delta[n]+7(-2)^{n-1}u[n-1]\tag{3}$$