BĐT
$\Leftrightarrow \left ( 1+xy \right )[1+ x^{2} +1+ y^{2}]\geq 2\left (1+ x^{2}\right ) \left (1+ y^{2} \right )$
$\Leftrightarrow \left ( 1+xy \right )[2 +x^{2}+y^{2}]\geq 2+2x^2+2y^2+2x^2y^2$
$\Leftrightarrow 2+x^2+y^2+xy\left ( 2+x^{2}+y^{2} \right )\geq 2+2x^2+2y^2+2x^2y^2$
$\Leftrightarrow xy\left ( x^{2}+y^{2}-2xy \right )-\left ( x^{2}+y^{2}-2xy \right )\geq 0$
$\Leftrightarrow \left ( x-y \right )^{2}\left ( xy-1 \right )\geq 0.$
Đúng vì $x,y\geq 1.$