90's kid

$Let~ y=x+\dfrac{4}{x}.~~$ Differentiating and equating to zero we get x = 2 or -2. Since factorials are for + tive integers, we have $\\a = 2 + \dfrac{4}{2} = 4. \\So~ (a!-2a) =~~ (4!-2*4) ~~= (24-8) = 16$

By AM-GM inequality you will get:

x+ $\frac{4}{x}$ $\geq$ 2 $\sqrt\frac{4x}{x}$

x+ $\frac{4}{x}$ $\geq$ 4

Substituting the value of $a=4$ in the equation gives: $(4!-2\times4)=(24-8)= \boxed{16}$