Homework Help: Why does E + dA/dt = - ∇V?

1. The problem statement, all variables and given/known data
This isn't necessarily a question but something I found in my notes that I didn't understand.
It says that from B = ∇ x A one can plug this into Faraday-Lenz's law ∇ x E = d(B)/dt and it leads to the equation:
∇ x (E + d(A)/dt)=0.
From this is says "Using Helmholtz theorem, one can represent the curl free field as the gradient of a scalar potential."

2. Relevant equations
B = ∇ x A
∇ x E = d(B)/dt
∇ x (E + d(A)/dt)=0

3. The attempt at a solution
I know that in electrostatics the curl of the electric field is zero and hence there is a scalar potential associated such that E = -∇ V but I don't see the jump for the case involving the vector potential.

Any guidance/explanation would be greatly appreciated.