$\begin{align}  & f(x)=\left| x \right|\,\,\,\,,\,\,\,\,\,g(x)={{x}^{2}}+2x+1={{(x+1)}^{2}} \\  & (fog)(1-\sqrt{2})-(gof)(1-\sqrt{2}) \\  & =f(g(1-\sqrt{2}))-g(f(1-\sqrt{2})) \\  & =f({{(1-\sqrt{2}+1)}^{2}})-g(\overbrace{\left| 1-\sqrt{2} \right|}^{-}) \\  & =f({{(2-\sqrt{2})}^{2}})-g(\sqrt{2}-1) \\  & =\left| 4+2-4\sqrt{2} \right|-{{(\sqrt{2}-1+1)}^{2}} \\  & =\overbrace{\left| 6-4\sqrt{2} \right|}^{+}-2=6-4\sqrt{2}-2=4(1-\sqrt{2}) \\ \end{align}$