[tex]f(x)=x^2-3\\\\f(f(x)) = f(x^2-3)=(x^2-3)^2-3=(x^2)^2-2\cdot x^2\cdot3+3^2-3=\\\\=x^4-6x^2+9-3 = \boxed{x^4-6x^2+6}[/tex]
