\(a\text{)}.\:\dfrac{1}{x+\sqrt{x}}+\dfrac{2\sqrt{x}}{x-1}-\dfrac{1}{x-\sqrt{x}}\\ =\dfrac{x-\sqrt{x}-x-\sqrt{x}}{\left(x+\sqrt{x}\right)\left(x-\sqrt{x}\right)}+\dfrac{2\sqrt{x}}{x-1}\\ =\dfrac{-2\sqrt{x}}{x\left(x-1\right)}+\dfrac{2\sqrt{x}}{x-1}=\dfrac{-2\sqrt{x}}{x\left(x-1\right)}+\dfrac{2x\sqrt{x}}{x\left(x-1\right)}\\ =\dfrac{2\sqrt{x}\left(x-1\right)}{x\left(x-1\right)}=\dfrac{2\sqrt{x}}{x}=\dfrac{2}{\sqrt{x}}\)
\(b\text{)}.\: \left(\dfrac{1}{2\sqrt{a}-a}+\dfrac{1}{2\sqrt{a}+a}\right):\dfrac{\sqrt{a}+1}{a-2\sqrt{a}}\\ =\dfrac{4\sqrt{a}}{4a-a^2}:\dfrac{\sqrt{a}+1}{a-2\sqrt{a}}=\dfrac{4\sqrt{a}}{a\left(4-a\right)}.\dfrac{\sqrt{a}\left(\sqrt{a}-2\right)}{\sqrt{a}+1}\\ =\dfrac{4\left(\sqrt{a}-2\right)}{\left(4-a\right)\left(\sqrt{a}+1\right)}=\dfrac{-4\left(2-\sqrt{a}\right)}{\left(2+\sqrt{a}\right)\left(2-\sqrt{a}\right)\left(\sqrt{a}+1\right)}\\ =-\dfrac{4}{\left(2+\sqrt{a}\right)\left(\sqrt{a}+1\right)}\)
\(c\text{)}.\:\dfrac{1}{2+2\sqrt{a}}+\dfrac{1}{2-2\sqrt{a}}-\dfrac{a^2+1}{1-a^2}\\ =\dfrac{4}{4-4a}-\dfrac{a^2+1}{1-a^2}=\dfrac{1}{1-a}-\dfrac{a^2+1}{\left(1-a\right)\left(1+a\right)}\\ =\dfrac{\left(1-a\right)\left(1+a\right)-a^2-1}{\left(1-a\right)^2\left(1+a\right)}=\dfrac{-2a^2}{\left(1-a\right)^2\left(1+a\right)}\)
