a: \(A=\dfrac{a+2\sqrt{a}}{\sqrt{a}+2}+\dfrac{a-4}{\sqrt{a}-2}\)
\(=\dfrac{\sqrt{a}\left(\sqrt{a}+2\right)}{\sqrt{a}+2}+\dfrac{\left(\sqrt[]{a}+2\right)\left(\sqrt{a}-2\right)}{\sqrt[]{a}-2}\)
\(=\sqrt{a}+\sqrt{a}+2=2\sqrt{a}+2\)
b: \(B=\dfrac{x+4\sqrt{x}+4}{\sqrt{x}+2}+\dfrac{x}{\sqrt{x}}\)
\(=\dfrac{\left(\sqrt{x}+2\right)^2}{\sqrt{x}+2}+\sqrt{x}\)
\(=\sqrt{x}+2+\sqrt{x}=2\sqrt{x}+2\)
c: \(C=\left(\dfrac{x+2\sqrt{x}}{\sqrt{x}+2}-5\right)\cdot\left(\dfrac{x-\sqrt{x}}{\sqrt{x}-1}+5\right)\)
\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\sqrt{x}+2}-5\right)\left(\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}-1}+5\right)\)
\(=\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)=x-25\)
d: \(D=\left(\dfrac{\sqrt{x}+1}{x-1}-\dfrac{\sqrt{x}}{x+\sqrt{x}}\right):\dfrac{1}{\sqrt{x}-1}\)
\(=\left(\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)\cdot\left(\sqrt{x}-1\right)\)
\(=\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}+1}\right)\cdot\left(\sqrt{x}-1\right)\)
\(=\dfrac{\sqrt{x}+1-\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\left(\sqrt{x}-1\right)=\dfrac{2}{\sqrt{x}+1}\)
