\(A=\left(\dfrac{2}{\sqrt{x}+1}+\dfrac{2+\sqrt{x}}{x-\sqrt{x}-2}\right):\left(1-\dfrac{5}{4-x}\right)\)
\(=\dfrac{2\sqrt{x}-4+\sqrt{x}+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}:\dfrac{x-4+5}{x-4}\)
\(=\dfrac{\sqrt{x}-2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}\cdot\dfrac{x-4}{x+1}\)
\(=\dfrac{x-4}{\left(x+1\right)\left(\sqrt{x}+1\right)}\)
\(A=\left(\dfrac{2}{\sqrt{x}+1}-\dfrac{2+5}{2+\sqrt{x}-x}\right):\left(1-\dfrac{3}{4-x}\right)\left(x\ge0;x\ne4\right)\\ A=\dfrac{4-2\sqrt{x}-3}{\left(2-\sqrt{x}\right)\left(\sqrt{x}+1\right)}:\dfrac{1-x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\\ A=\dfrac{2\sqrt{x}-1}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\cdot\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{1-x}=\dfrac{2\sqrt{x}-1}{1-x}\)
\(x=\sqrt{29+12\sqrt{3-2\sqrt{2}}}=\sqrt{29+12\left(\sqrt{2}-1\right)}\\ x=\sqrt{17+12\sqrt{2}}=3+2\sqrt{2}\\ \Leftrightarrow A=\dfrac{2\sqrt{3+2\sqrt{2}}-1}{1-3+2\sqrt{2}}=\dfrac{2\left(\sqrt{2}+1\right)-1}{2\sqrt{2}-2}=\dfrac{2\sqrt{2}+1}{2\sqrt{2}-2}\\ A=\dfrac{\left(2\sqrt{2}+1\right)\left(2\sqrt{2}+2\right)}{4}=\dfrac{10+6\sqrt{2}}{4}=\dfrac{5+3\sqrt{2}}{2}\)