Đáp án:

$1)\\ a)9-x\\ b)\dfrac{2}{x+\sqrt{x}+1}\\ c)\dfrac{\sqrt{x}+3}{\sqrt{x}+1}\\ d)\dfrac{\sqrt{x}-2}{\sqrt{x}-1}$

Giải thích các bước giải:

$1)\\ a)\left(3-\dfrac{x-\sqrt{x}}{\sqrt{x}-1}\right)\left(3+\dfrac{x+\sqrt{x}}{\sqrt{x}+1}\right)\\ =\left(3-\dfrac{\sqrt{x}(\sqrt{x}-1)}{\sqrt{x}-1}\right)\left(3+\dfrac{\sqrt{x}(\sqrt{x}+1)}{\sqrt{x}+1}\right)\\ =\left(3-\sqrt{x}\right)\left(3+\sqrt{x}\right)\\ =9-x\\ b)\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right).\dfrac{2}{\sqrt{x}-1}\\ =\left(\dfrac{x+2}{(\sqrt{x}-1)(x+\sqrt{x}+1)}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right).\dfrac{2}{\sqrt{x}-1}\\ =\left(\dfrac{x+2}{(\sqrt{x}-1)(x+\sqrt{x}+1)}+\dfrac{\sqrt{x}(\sqrt{x}-1)}{(\sqrt{x}-1)(x+\sqrt{x}+1)}-\dfrac{x+\sqrt{x}+1}{(\sqrt{x}-1)(x+\sqrt{x}+1)}\right).\dfrac{2}{\sqrt{x}-1}\\ =\dfrac{x+2+\sqrt{x}(\sqrt{x}-1)-(x+\sqrt{x}+1)}{(\sqrt{x}-1)(x+\sqrt{x}+1)}.\dfrac{2}{\sqrt{x}-1}\\ =\dfrac{x-2\sqrt{x}+1}{(\sqrt{x}-1)(x+\sqrt{x}+1)}.\dfrac{2}{\sqrt{x}-1}\\ =\dfrac{(\sqrt{x}-1)^2}{(\sqrt{x}-1)(x+\sqrt{x}+1)}.\dfrac{2}{\sqrt{x}-1}\\ =\dfrac{2}{x+\sqrt{x}+1}\\ c)\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}+2}\right):\left(\dfrac{3}{\sqrt{x}+3}-\dfrac{1}{\sqrt{x}+2}\right)\\ =\left(\dfrac{\sqrt{x}+2}{(\sqrt{x}+1)(\sqrt{x}+2)}+\dfrac{\sqrt{x}+1}{(\sqrt{x}+2)(\sqrt{x}+1)}\right):\left(\dfrac{3(\sqrt{x}+2)}{(\sqrt{x}+3)(\sqrt{x}+2)}-\dfrac{\sqrt{x}+3}{(\sqrt{x}+2)(\sqrt{x}+3)}\right)\\ =\dfrac{\sqrt{x}+2+\sqrt{x}+1}{(\sqrt{x}+2)(\sqrt{x}+1)}:\dfrac{3(\sqrt{x}+2)-(\sqrt{x}+3)}{(\sqrt{x}+2)(\sqrt{x}+3)}\\ =\dfrac{2\sqrt{x}+3}{(\sqrt{x}+2)(\sqrt{x}+1)}:\dfrac{2\sqrt{x}+3}{(\sqrt{x}+2)(\sqrt{x}+3)}\\ =\dfrac{2\sqrt{x}+3}{(\sqrt{x}+2)(\sqrt{x}+1)}.\dfrac{(\sqrt{x}+2)(\sqrt{x}+3)}{2\sqrt{x}+3}\\ =\dfrac{\sqrt{x}+3}{\sqrt{x}+1}\\ d)\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{\sqrt{x}+1}{1-\sqrt{x}}-\dfrac{x+\sqrt{x}+4}{x-1}\\ =\dfrac{\sqrt{x}-1}{\sqrt{x}+1}+\dfrac{\sqrt{x}+1}{\sqrt{x}-1}-\dfrac{x+\sqrt{x}+4}{(\sqrt{x}-1)(\sqrt{x}+1)}\\ =\dfrac{(\sqrt{x}-1)^2+(\sqrt{x}+1)^2-(x+\sqrt{x}+4)}{(\sqrt{x}-1)(\sqrt{x}+1)}\\ =\dfrac{x-\sqrt{x}-2}{(\sqrt{x}-1)(\sqrt{x}+1)}\\ =\dfrac{(\sqrt{x}-2)(\sqrt{x}+1)}{(\sqrt{x}-1)(\sqrt{x}+1)}\\ =\dfrac{\sqrt{x}-2}{\sqrt{x}-1}$