\(\sqrt{\sqrt{2}-1}+\sqrt{\sqrt{2}+1}-\sqrt{2\sqrt{2}+2}=\sqrt{\sqrt{2}-1}+\sqrt{\sqrt{2}+1}-\sqrt{2}.\sqrt{\sqrt{2}+1}=\sqrt{\sqrt{2}-1}+\sqrt{\sqrt{2}+1}\left(1-\sqrt{2}\right)=\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}+1}\left(\sqrt{2}-1\right)=\sqrt{\sqrt{2}-1}-\sqrt{\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}.\sqrt{\sqrt{2}-1}=\sqrt{\sqrt{2}-1}-\sqrt{1}.\sqrt{\sqrt{2}-1}=\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}=0\)