i get everything until the red line, why is the next step like that? and where did the part on top of the green line come from?
$\displaystyle |x|^{2} - 2|x||y| + |y|^{2} \geq 0$
$\displaystyle \Rightarrow |x|^{2} + |y|^{2} \geq 2|x| |y|$ ..........................................(Moved 2|x||y| to right side)
$\displaystyle \Rightarrow |x|^{2} + |y|^{2} {\color{red} + |x|^{2} + |y|^{2}} \geq 2|x| |y| {\color{red} + |x|^{2} + |y|^{2}}$..........(Added $\displaystyle |x|^2 + |y|^2$ to both sides)
$\displaystyle \Rightarrow 2|x|^{2} + 2|y|^{2} \geq |x|^{2} + 2|x||y| + |y|^{2}$......................(Simplified)