so "any x that satisfies the equation " simply means and .

If you put into the formula you get [tex](-1+\sqrt{3})^3+ (-1+ \sqrt{3})^2+ A(-1+\sqrt{3})+ 1= 6\sqrt{3}- 10+ 4- 2\sqrt{3}- A- A\sqrt{3}+ 1= -5- A+ (4+A)\sqrt{3}[/itex].

Do you see what A must be in order that P be a rational number?