On a problem of matrix equation

On a problem of matrix equation

The problem goes as follows:
For given continuous $A:[0,1]\to \mathbb{C}^{n\times n}$ with all the real
part of its eigenvalues is positive, then show there must exist some
continuous $B:[0,1] \to \mathbb{C}$ such that $$(B(t)+I)(B(t)-2I)=A(t)$$
I suppose the continuous here means the continuity derived from canonical
Euclidean topology of $\mathbb{C}^{n^2}\simeq\mathbb{R}^{2n^2}$, but
that's not quite the first problem we should solve. I get totally no idea
of how to do it. It seems unrealistic to expect that $B(\cdot)$ can be
defined pointwise since there is no any condition on the diagolizability
of $A$, and since all it asks is about existence theory, explicit
construction may be out of question for the first place.