4:
\(P=\left(\dfrac{4x}{4-x}+\dfrac{2+\sqrt{x}}{2-\sqrt{x}}-\dfrac{2-\sqrt{x}}{2+\sqrt{x}}\right):\dfrac{\sqrt{x}+3}{2-\sqrt{x}}\)
\(=\dfrac{4x+\left(2+\sqrt{x}\right)^2-\left(2-\sqrt{x}\right)^2}{\left(2-\sqrt{x}\right)\left(2+\sqrt{x}\right)}\cdot\dfrac{2-\sqrt{x}}{\sqrt{x}+3}\)
\(=\dfrac{4x+x+4\sqrt{x}+4-x+4\sqrt{x}-4}{\left(\sqrt{x}+2\right)}\cdot\dfrac{1}{\sqrt{x}+3}\)
\(=\dfrac{4x+8\sqrt{x}}{\sqrt{x}+2}\cdot\dfrac{1}{\sqrt{x}+3}=\dfrac{4\sqrt{x}}{\sqrt{x}+3}\)
5:
\(A=\left(\dfrac{x+2\sqrt{x}}{x-2\sqrt{x}}+\dfrac{\sqrt{x}}{\sqrt{x}-2}\right)\cdot\dfrac{1}{\sqrt{x}+1}\)
\(=\left(\dfrac{\sqrt{x}+2}{\sqrt{x}-2}+\dfrac{\sqrt{x}}{\sqrt{x}-2}\right)\cdot\dfrac{1}{\sqrt{x}+1}\)
\(=\dfrac{2\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}=\dfrac{2}{\sqrt{x}-2}\)
