My question is how to use Mathematica to check if one expression can be transformed into another:

Example 1: $ ax^2+bx+c$ into $ K\cdot(x+\alpha)^2+\beta$ , and then show me $ K(a,b,c)$ and so on.

Example 2: Say I have the Maxwell equations:

$ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$ , $ \nabla \times \mathbf{E} = -\dot {\mathbf{B}}$ , $ \nabla \cdot \mathbf{B} = 0$ , $ \nabla \times \mathbf{B} = \mu_0 (\varepsilon_0 \mathbf{j} + \dot{\mathbf{E}})$ .

Can I follow from this an expression of the form $ \alpha \ddot {\mathbf{E}}+\beta \mathbf{E}=0$ ?

Example 3: $ c_1 \cos(\omega t) + c_2 \sin(\omega t)$ into $ A cos (\omega t + \delta)$ . How do I mark the dependencies, so that $ A$ is $ A(c_1, c_2)$ but not $ A(t)$ ?