\(\left(\dfrac{1}{x}+x-2\right):\left(\dfrac{1}{x^2-x}+1-\dfrac{3}{x-1}\right)\)

\(=\dfrac{x^2-2x+1}{x}:\dfrac{1+x^2-x-3x}{x\left(x-1\right)}\)

\(=\dfrac{\left(x-1\right)^2}{x}\cdot\dfrac{x\left(x-1\right)}{x^2-4x+1}=\dfrac{\left(x-1\right)^3}{x^2-4x+1}\)

a. \(\left(2x+1\right)^2-4x\left(x-1\right)=4x^2+4x+1-4x^2+4x=8x+1\)

b. \(\left(x-2\right)\left(x+2\right)-\left(x-1\right)^2=x^2-4-x^2+2x-1=2x-5\)

a)\((2x+1)^2-4x(x-1)=4x^2+4x+1-4x^2+4x\)

=\(8x+ 1\)

b)\((X-2)(x+2)-(x-1)^2=X^2-4-(x^2-2x+1)\)

=\(X^2-4-x^2+2x-1=2x-5\)

a: \(=\dfrac{x^2-2x+1}{x}:\dfrac{x-1-3x^2+3x-3}{\left(x-1\right)\left(x^2-x+1\right)}\)

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

\(=\dfrac{\left(x-1\right)^3\cdot\left(x^2-x+1\right)}{-2x\left(x^2-2x+2\right)}\)

b: \(=\left[\dfrac{x^2-2x+1}{x^2+x+1}+\dfrac{2x^2-4x+1}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{1}{x-1}\right]:\dfrac{2}{x^2+1}\)

\(=\dfrac{x^3-3x^2+3x+1+2x^2-4x+1+x^2+x+1}{\left(x-1\right)\left(x^2+x+1\right)}\cdot\dfrac{x^2+1}{2}\)

\(=\dfrac{x^3+3}{\left(x-1\right)\left(x^2+x+1\right)}\cdot\dfrac{x^2+1}{2}\)

\(a,=2x^2-7xy-30y^2+30x^2+2xy=32x^2-5xy-30y^2\\ b,=x^2-10x+25+2x^2-8=3x^2-10x+17\\ c,=x^3+8-x^3+5=13\\ d,=x^3-x^2+x-x^2+x-1+x^2-1=x^3-x^2+2x-2\)

\(a,\left(x-2\right)\left(x+3\right)-x\left(x-5\right)=x^2-2x+3x-6-x^2+5x=6x-6\)

\(b,\dfrac{1}{x-2}+\dfrac{-2}{x+2}+\dfrac{2x-8}{x^2-4}=\dfrac{x+2}{\left(x-2\right)\left(x+2\right)}-\dfrac{2\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}+\dfrac{2x-8}{\left(x+2\right)\left(x-2\right)}=\dfrac{x+2}{\left(x-2\right)\left(x+2\right)}-\dfrac{2x-4}{\left(x+2\right)\left(x-2\right)}+\dfrac{2x-8}{\left(x+2\right)\left(x-2\right)}=\dfrac{x+2-2x+4+2x-8}{\left(x+2\right)\left(x-2\right)}=\dfrac{x-2}{\left(x+2\right)\left(x-2\right)}=\dfrac{1}{x+2}\)

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

\(\dfrac{xy\left(x^2+y^2\right)}{xy\left(x^3\right)}.\dfrac{1}{x^2+y^2}=\dfrac{1}{x^3}\)

a: \(=\left(x+2\right)^2\cdot\dfrac{2x-1}{3\left(x+2\right)}=\dfrac{\left(x+2\right)\left(2x-1\right)}{3}\)

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

a: \(\frac{4\left(x+3\right)}{3x^2-x}:\frac{x^2+3x}{1-3x}\)

\(=\frac{4\left(x+3\right)}{x\left(3x-1\right)}\cdot\frac{-\left(3x-1\right)}{x\left(x+3\right)}=\frac{-4}{x^2}\)

b: \(\frac{x+1}{x^2-2x-8}\cdot\frac{4-x}{x^2+x}\)

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

c: \(\frac{9x+5}{2\left(x-1\right)\left(x+3\right)^2}-\frac{5x-7}{2\left(x-1\right)\left(x+3\right)^2}\)

\(=\frac{9x+5-5x+7}{2\left(x-1\right)\left(x+3\right)^2}=\frac{4x+12}{2\left(x-1\right)\left(x+3\right)^2}\)

\(=\frac{4\left(x+3\right)}{2\left(x-1\right)\left(x+3\right)^2}=\frac{2}{\left(x-1\right)\left(x+3\right)}\)

d: \(\frac{18}{\left(x-3\right)\left(x^2-9\right)}-\frac{3}{x^2-6x+9}-\frac{x}{x^2-9}\)

\(=\frac{18}{\left(x+3\right)\left(x-3\right)^2}-\frac{3}{\left(x-3\right)^2}-\frac{x}{\left(x-3\right)\left(x+3\right)}\)

\(=\frac{18-3\left(x+3\right)-x\left(x-3\right)}{\left(x-3\right)^2\cdot\left(x+3\right)}=\frac{18-3x-9-x^2+3x}{\left(x-3\right)^2\cdot\left(x+3\right)}=\frac{-x^2+9}{\left(x-3\right)^2\left(x+3\right)}=\frac{-1}{x-3}\)

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

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

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

\(=\frac{3x-4}{\left(x+1\right)\left(x-1\right)\left(x^2-x+1\right)}\)

a: \(x^2\left(x-2x^3\right)=x^2\cdot x-x^2\cdot2x^3=x^3-2x^5\)

b: \(\left(x^2+1\right)\cdot5x=x^2\cdot5x+1\cdot5x=5x^3+5x\)

c: \(\left(x-2\right)\left(x^2+2x+4\right)\)

\(=x^3+2x^2+4x-2x^2-4x-8=x^3-8\)

d: \(\left(x-2\right)\left(x^2+2x+4\right)\)

\(=x^3+2x^2+4x-2x^2-4x-8=x^3-8\)

e: \(\left(x^2-1\right)\left(x^2+1\right)=\left(x^2\right)^2-1^2=x^4-1\)

f: (2x-1)(3x+2)(3-x)

\(=\left(6x^2+4x-3x-2\right)\left(3-x\right)\)

\(=\left(6x^2+x-2\right)\left(-x+3\right)\)

\(=-6x^3+18x^2-x^2+3x+2x-6=-6x^3+17x^2+5x-6\)

\(a,\left(2x-5\right)\left(5-x\right)=5\left(2x-5\right)-x\left(2x-5\right)=10x-25-2x^2+5x=15x-2x^2-25\\ b,\dfrac{1}{3x-2}-\dfrac{1}{3x+2}=\dfrac{3x+2-3x+2}{\left(3x-2\right)\left(3x+2\right)}=\dfrac{4}{\left(3x-2\right)\left(3x+2\right)}\)

\(c,\dfrac{3}{x-3}-\dfrac{6x}{x^2-9}+\dfrac{x}{x+3}=\dfrac{3\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\dfrac{6x}{\left(x-3\right)\left(x+3\right)}+\dfrac{x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}=\dfrac{3x+9-6x+x^2-3x}{\left(x-3\right)\left(x+3\right)}=\dfrac{x^2-6x+9}{\left(x-3\right)\left(x+3\right)}=\dfrac{\left(x-3\right)^2}{\left(x-3\right)\left(x+3\right)}=\dfrac{x-3}{x+3}\)