\(\left(\dfrac{x+\sqrt{x}+1}{x+\sqrt{x}-2}+\dfrac{1}{\sqrt{x}-1}+\dfrac{1}{\sqrt{x}+2}\right):\dfrac{1}{x+1}\\ =\left[\dfrac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}+\dfrac{1}{\sqrt{x}-1}+\dfrac{1}{\sqrt{x}+2}\right]:\dfrac{1}{x+1}\\ =\left[\dfrac{x+\sqrt{x}+1+\sqrt{x}+2+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\right]:\dfrac{1}{x+1}\\ =\left[\dfrac{x+3\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\right]\cdot\left(x+1\right)\\ =\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x} +1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}\cdot\left(x+1\right)\\ =\dfrac{\left(\sqrt{x}+1\right)\cdot\left(x+1\right)}{\sqrt{x}-1}\\ =\dfrac{\left(\sqrt{x}+1\right)\cdot\left(x+1\right)}{\sqrt{x}-1}\)
