\(\left(x+\sqrt{x^2+2015}\right)\left(y+\sqrt{y^2+2015}\right)=2015\)

\(\Leftrightarrow\frac{2015}{\sqrt{x^2+2015}-x}\left(y+\sqrt{y^2+2015}\right)=2015\)

\(\Leftrightarrow\sqrt{x^2+2015}-x=y+\sqrt{y^2+2015}\left(1\right)\)

Tương tự : \(x+\sqrt{x^2+2015}=\sqrt{y^2+2015}-y\left(2\right)\)

(1)+(2):

\(x+\sqrt{x^2+2015}+y+\sqrt{y^2+2015}=\sqrt{x^2+2015}+\sqrt{y^2+2015}-x-y\)

\(\Leftrightarrow2\left(x+y\right)=0\Leftrightarrow x+y=0\)

\(x^4-16x^2+32=0\Leftrightarrow x^2=8+4\sqrt{2}\text{ hoặc }x^2=8-4\sqrt{2}\)

\(a=\sqrt{2+\sqrt{\frac{4+2\sqrt{3}}{2}}}-\sqrt{6-3\sqrt{\frac{4+2\sqrt{3}}{2}}}\)\(=\sqrt{2+\frac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{2}}}-\sqrt{6-3\frac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{2}}}\)

\(=\sqrt{2+\frac{\sqrt{3}+1}{\sqrt{2}}}-\sqrt{6-3\frac{\sqrt{3}+1}{\sqrt{2}}}=\sqrt{\frac{4+\sqrt{6}+\sqrt{2}}{2}}-\sqrt{3}\sqrt{\frac{4-\sqrt{6}-\sqrt{2}}{2}}\)

\(a^2=\frac{4+\sqrt{6}+\sqrt{2}}{2}+3.\frac{4-\sqrt{6}-\sqrt{2}}{2}-2\sqrt{3}\sqrt{\frac{\left(4+\sqrt{6}+\sqrt{2}\right)\left(4-\sqrt{6}-\sqrt{2}\right)}{2.2}}\)

\(=8-\left(\sqrt{6}+\sqrt{2}\right)-2\sqrt{3}.\frac{1}{2}.\sqrt{4^2-\left(\sqrt{6}+\sqrt{2}\right)^2}\)

\(=8-\sqrt{6}-\sqrt{2}-\sqrt{3}\sqrt{8-4\sqrt{3}}\)

\(=8-\sqrt{2}-\sqrt{6}-\sqrt{\left(3\sqrt{2}-\sqrt{6}\right)^2}\)

\(=8-\sqrt{2}-\sqrt{6}-\left(3\sqrt{2}-\sqrt{6}\right)\)

\(=8-4\sqrt{2}\)

\(\Rightarrow a\text{ là nghiệm phương trình }x^4-16x^2+32=0\)

\(x^2=2+\sqrt{2+\sqrt{3}}+6-3\sqrt{2+\sqrt{3}}-2.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{6-3\sqrt{2+\sqrt{3}}}\)

\(x^2=8-2\sqrt{2+\sqrt{3}}-2.\sqrt{3.\left(2+\sqrt{2+\sqrt{3}}\right).\left(2-\sqrt{2+\sqrt{3}}\right)}\)

\(x^2=8-2\sqrt{2+\sqrt{3}}-2.\sqrt{3.\left(4-\left(2+\sqrt{3}\right)\right)}=8-2\sqrt{2+\sqrt{3}}-2.\sqrt{3.\left(2-\sqrt{3}\right)}\)

\(x^2=8-\sqrt{2}\sqrt{4+2.\sqrt{3}}-\sqrt{6}.\sqrt{4-2.\sqrt{3}}=8-\sqrt{2}.\sqrt{\left(1+\sqrt{3}\right)^2}-\sqrt{6}.\sqrt{\left(\sqrt{3}-1\right)^2}\)

\(x^2=8-\sqrt{2}.\left(1+\sqrt{3}\right)-\sqrt{6}.\left(\sqrt{3}-1\right)=8-\sqrt{2}-\sqrt{6}-3\sqrt{2}+\sqrt{6}=8-4\sqrt{2}\)

=> \(x^4=\left(x^2\right)^2=\left(8-4\sqrt{2}\right)^2=\left(4\sqrt{2}\right)^2.\left(\sqrt{2}-1\right)^2=32.\left(2-2\sqrt{2}+1\right)=96-64\sqrt{2}\)

=> \(x^4-16x^2+32=96-64\sqrt{2}-16.\left(8-4\sqrt{2}\right)+32=\left(96-96\right)-64\sqrt{2}+64\sqrt{2}=0\)

=> đpcm

9y^2-x^2-4x-4

=9y²-(x²+4x+4)

=9y²-(x+2)²

=(3y-x-2)(3y+x+2)