Find the domain of F (x-2) = 4-3x-x^2

The domain of any polynomial is "the set of all real numbers."

I think you may have mistyped your function, meaning F (x) = 4-3x-x^2. But either way, F (x-2) = 4-3x-x^2 and F (x) are polynomials, and so their domains are "the set of all real numbers."