\(P=\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3}{14-8\sqrt{3}}}-3\sqrt{2\left(1-\sqrt{-2\sqrt{3}+4}\right)+2\sqrt{4+2\sqrt{3}}}=\left(12-6\sqrt{3}\right)\sqrt{\dfrac{3\left(14+8\sqrt{3}\right)}{\left(14-8\sqrt{3}\right)\left(14+8\sqrt{3}\right)}}-3\sqrt{2\left(1-\sqrt{3-2\sqrt{3}+1}\right)+2\sqrt{3+2\sqrt{3}+1}}\)\(=\left(12-6\sqrt{3}\right)\sqrt{\dfrac{42+24\sqrt{3}}{4}}-3\sqrt{2\left(1-\sqrt{\left(\sqrt{3}-1\right)^2}\right)+2\sqrt{\left(\sqrt{3}+1\right)^2}}=\dfrac{6\sqrt{\left(2-\sqrt{3}\right)^2\left(42+24\sqrt{3}\right)}}{2}-3\sqrt{2\left(1-\sqrt{3}+1\right)+2\left(\sqrt{3}+1\right)}=3\sqrt{\left(7-4\sqrt{3}\right)\left(42+24\sqrt{3}\right)}-3\sqrt{2-2\sqrt{3}+2+2\sqrt{3}+2}=3\sqrt{294+168\sqrt{3}-168\sqrt{3}-288}-3\sqrt{6}=3\sqrt{6}-3\sqrt{6}=0\)