Ta có: \(\frac{x^2-xy}{x^2y+y^3}-\frac{2x^2}{y^3-xy_{}^2+x^2y-x^3}\)
\(=\frac{x\left(x-y\right)}{y\left(x^2+y^2\right)}-\frac{2x^2}{y^2\left(y-x\right)+x^2\left(y-x\right)}\)
\(=\frac{x\left(x-y\right)}{y\left(x^2+y^2\right)}+\frac{2x^2}{\left(x-y\right)\left(x^2+y^2\right)}=\frac{x\left(x-y\right)\left(x-y\right)+2x^2y}{y\left(x-y\right)\left(x^2+y^2\right)}\)
\(=\frac{x\left(x^2-2xy+y^2\right)+2x^2y}{y\left(x-y\right)\left(x^2+y^2\right)}=\frac{x\left(x^2+y^2\right)}{y\left(x-y\right)\left(x^2+y^2\right)}\)
\(=\frac{x}{y\left(x-y\right)}\)
Ta có: \(1-\frac{y-1}{x}-\frac{y}{x^2}\)
\(=\frac{x^2-x\left(y-1\right)-y}{x^2}\)
\(=\frac{x^2-xy+x-y}{x^2}=\frac{\left(x-y\right)\left(x+1\right)}{x^2}\)
Ta có; \(A=\left(\frac{x^2-xy}{x^2y+y^3}-\frac{2x^2}{y^3-xy_{}^2+x^2y-x^3}\right)\cdot\left(1-\frac{y-1}{x}-\frac{y}{x^2}\right)\)
\(=\frac{x}{y\left(x-y\right)}\cdot\frac{\left(x-y\right)\left(x+1\right)}{x^2}=\frac{x+1}{xy}\)
Ta có: \(4x^4-20x^3+13x^2+30x+9\)
\(=4x^4+2x^3-22x^3-11x^2+24x^2+12x+18x+9\)
\(=2x^3\left(2x+1\right)-11x^2\left(2x+1\right)+12x\left(2x+1\right)+9\left(2x+1\right)\)
\(=\left(2x+1\right)\left(2x^3-11x^2+12x+9\right)\)
\(=\left(2x+1\right)\left(2x^3+x^2-12x^2-6x+18x+9\right)\)
\(=\left(2x+1\right)\left(2x+1\right)\left(x^2-6x+9\right)=\left(2x+1\right)^2\cdot\left(x-3\right)^2\)
Ta có: \(B=\frac{4x^4-20x^3+13x^2+30x+9}{\left(4x^2-1\right)^2}\)
\(=\frac{\left(2x+1\right)^2\cdot\left(x-3\right)^2}{\left(2x+1\right)^2\cdot\left(2x-1\right)^2}=\frac{\left(x-3\right)^2}{\left(2x-1\right)^2}\)
