a: \(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}\)

\(=\frac{2x}{x\left(x+2y\right)}+\frac{y}{y\left(x-2y\right)}+\frac{4}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2}{x+2y}+\frac{1}{x-2y}+\frac{4}{\left(x-2y\right)\left(x+2y\right)}=\frac{2\left(x-2y\right)+x+2y+4}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2x-4y+x+2y+4}{\left(x-2y\right)\left(x+2y\right)}=\frac{3x-2y+4}{\left(x-2y\right)\left(x+2y\right)}\)

b: \(\frac{2}{x+2}+\frac{4}{x-2}+\frac{5x+2}{4-x^2}\)

\(=\frac{2\left(x-2\right)+4\left(x+2\right)-5x-2}{\left(x-2\right)\left(x+2\right)}\)

\(=\frac{2x-4+4x+8-5x-2}{\left(x-2\right)\left(x+2\right)}=\frac{x+2}{\left(x-2\right)\left(x+2\right)}=\frac{1}{x-2}\)

c: \(\frac{x}{x-2y}+\frac{x}{x+2y}-\frac{4xy}{4y^2-x^2}\)

\(=\frac{x\left(x+2y\right)+x\left(x-2y\right)+4xy}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{x^2+2xy+x^2-2xy+4xy}{\left(x-2y\right)\left(x+2y\right)}=\frac{2x^2+4xy}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2x\left(x+2y\right)}{\left(x+2y\right)\left(x-2y\right)}=\frac{2x}{x-2y}\)

d: \(\frac{3x^2-x}{x-1}+\frac{x+2}{1-x}+\frac{3-2x^2}{x-1}\)

\(=\frac{3x^2-x-x-2+3-2x^2}{x-1}=\frac{x^2-2x+1}{x-1}\)

\(=\frac{\left(x-1\right)^2}{x-1}=x-1\)

Ta có:  \(2\left(x-\frac{1}{2}\right)+3\left(-1+\frac{x}{3}\right)=x\left(\frac{2}{x}-1\right)\)

           \(2x-1+-3+\frac{3x}{3}=\frac{2x}{x}-x\)

           \(2x-1+-3+x=2-x\)

           \(\left(2x+x\right)+\left(-3\right)-1=2-x\)

           \(3x+\left(-4\right)=2-x\)

           \(3x+x=2-\left(-4\right)\)  

           \(4x=6\)

           \(x=6:4\)

           \(x=\frac{6}{4}=\frac{3}{2}\)

mik nha

\(P=\left(\dfrac{1}{x-1}-\dfrac{2x}{x^3-x^2+x-1}\right):\left(\dfrac{1-2x}{x+1}\right)\left(ĐKXĐ:x\ne0;x\ne\pm1\right)\)

\(=\left(\dfrac{1}{x-1}-\dfrac{2x}{x^2\left(x-1\right)+\left(x-1\right)}\right):\left(\dfrac{1-2x}{x+1}\right)\)

\(=\left(\dfrac{1}{x-1}-\dfrac{2x}{\left(x-1\right)\left(x^2+1\right)}\right):\left(\dfrac{1-2x}{x+1}\right)\)

\(=\left(\dfrac{x^2+1-2x}{\left(x-1\right)\left(x^2+1\right)}\right):\left(\dfrac{1-2x}{x+1}\right)\)

\(=\dfrac{\left(x-1\right)^2}{\left(x-1\right)\left(x^2+1\right)}:\dfrac{1-2x}{x+1}\)

\(=\dfrac{x-1}{x^2+1}:\dfrac{1-2x}{x+1}\)

\(=\dfrac{x-1}{x^2+1}.\dfrac{x+1}{1-2x}\)

\(=\dfrac{x^2-1}{\left(x^2+1\right)\left(1-2x\right)}\)

1: \(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\)

\(=\frac{\sqrt{x}\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}+3}{\sqrt{x}-1}\)

\(=\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}+3}{\sqrt{x}-1}=\frac{2\sqrt{x}+3}{\sqrt{x}-1}\)

\(\frac{x-1}{2x+\sqrt{x}-1}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2x+2\sqrt{x}-\sqrt{x}-1}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{2\sqrt{x}-1}\)

Ta có: \(B=\left(\frac{x\sqrt{x}+x+\sqrt{x}}{x\sqrt{x}-1}-\frac{\sqrt{x}+3}{1-\sqrt{x}}\right)\cdot\frac{x-1}{2x+\sqrt{x}-1}\)

\(=\frac{2\sqrt{x}+3}{\sqrt{x}-1}\cdot\frac{\sqrt{x}-1}{2\sqrt{x}-1}=\frac{2\sqrt{x}+3}{2\sqrt{x}-1}\)

2: \(\frac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\cdot\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)=x-1\)

Ta có: \(A=\frac{\left(\sqrt{x}-1\right)^2\left(\sqrt{x}+1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{3\sqrt{x}+1}{x-1}\)

\(=x-1-\frac{3\sqrt{x}+1}{x-1}=\frac{\left(x-1\right)^2-3\sqrt{x}-1}{x-1}=\frac{x^2-2x-3\sqrt{x}}{x-1}\)

\(=\frac{\sqrt{x}\left(x\sqrt{x}-2\sqrt{x}-3\right)}{x-1}\)

\(F=\left(\dfrac{2\sqrt{x}}{2\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right)\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}+\dfrac{3x}{x-2\sqrt{x}+1}\right)\left(x>0;x\ne1;x\ne\dfrac{1}{4}\right)\\ F=\dfrac{2x-2\sqrt{x}+1}{\sqrt{x}\left(2\sqrt{x}-1\right)}\cdot\dfrac{x-1+3x}{\left(\sqrt{x}-1\right)^2}\\ F=\dfrac{2x-2\sqrt{x}+1}{\sqrt{x}\left(2\sqrt{x}-1\right)}\cdot\dfrac{\left(2\sqrt{x}-1\right)\left(2\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)^2}\\ F=\dfrac{\left(2\sqrt{x}+1\right)\left(2x-2\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)^2}\)

a: Ta có: \(F=\left(\dfrac{2\sqrt{x}}{2\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right)\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}+\dfrac{3x}{x-2\sqrt{x}+1}\right)\)

\(=\dfrac{2x-2\sqrt{x}+1}{\sqrt{x}\left(2\sqrt{x}-1\right)}\cdot\dfrac{4x-1}{\left(\sqrt{x}-1\right)^2}\)

\(=\dfrac{\left(2x-2\sqrt{x}+1\right)\left(2\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)^2}\)