\(\)

\($$ x = \frac{23}{\sqrt{31+12\sqrt{3}}} \\ x = \frac{23}{\sqrt{(3\sqrt{3}+2)^2}} \\ x = \frac{23}{3\sqrt{3}+2} \\ x = \frac{23(3\sqrt{3}-2)}{(3\sqrt{3}+2)(3\sqrt{3}-2)} \\ x = \frac{23(3\sqrt{3}-2)}{27-4} \\ x = \frac{23(3\sqrt{3}-2)}{23} \\ x = 3\sqrt{3}-2 \\ \\ (x+2)^2 = (3\sqrt{3})^2 \\ x^2+4x+4 = 27 \\ x^2+4x-23 = 0 \\ \\ P = \frac{x^4+5x^3-20x^2-27x+30}{x^2+4x-21} \\ \\ x^2+4x-21 = (x^2+4x-23)+2 = 0+2 = 2 \\ \\ x^4+5x^3-20x^2-27x+30 = x^2(x^2+4x-23) + x(x^2+4x-23) - (x^2+4x-23) + 7 \\ = x^2(0) + x(0) - (0) + 7 \\ = 7 \\ \\ P = \frac{7}{2} $$ \)

\frac{23}{\sqrt{31+12\sqrt{3}}} x = \frac{23}{\sqrt{(3\sqrt{3}+2)^2}} x = \frac{23}{3\sqrt{3}+2} x = \frac{23(3\sqrt{3}-2)}{(3\sqrt{3}+2)(3\sqrt{3}-2)} x = \frac{23(3\sqrt{3}-2)}{27-4} x = \frac{23(3\sqrt{3}-2)}{23} x = 3\sqrt{3}-2 (x+2)^2 = (3\sqrt{3})^2 x^2+4x+4 = 27 x^2+4x-23 = 0 P = \frac{x^4+5x^3-20x^2-27x+30}{x^2+4x-21} x^2+4x-21 = x^2+4x-23+2 = 0+2 = 2 x^4+5x^3-20x^2-27x+30 = x^2(x^2+4x-23) + x(x^2+4x-23) - (x^2+4x-23) + 7 = x^2(0) + x(0) - (0) + 7 = 7 P = \frac{7}{2}