Rút gọn các phân thức:
a) \(\frac{\left(3x+2\right)^2-\left(x+2\right)^2}{x^3-x^2}=\frac{9x^2+12x+4-x^2-4x-4}{x^3-x^2}=\frac{8x^2+8x}{x^3-x^2}=\frac{8x\left(x+1\right)}{x^2\left(x-1\right)}=\frac{8\left(x+1\right)}{x-1}\)
b) \(\frac{x^4-1}{x^3+2x^2-x-2}=\frac{\left(x^2-1\right)\left(x^2+1\right)}{\left(x^3-x\right)+\left(2x^2-2\right)}=\frac{\left(x^2-1\right)\left(x^2+1\right)}{\left(x+2\right)\left(x^2-1\right)}=\frac{x^2+1}{x+2}\)
c) \(\frac{x^2+7x+12}{x^2+5x+6}=\frac{\left(x^2+3x\right)+\left(4x+12\right)}{\left(x^2+3x\right)+\left(2x+6\right)}=\frac{\left(x+3\right)\left(x+4\right)}{\left(x++3\right)\left(x+2\right)}=\frac{x+4}{x+2}\)
d) \(\frac{x^{10}-x^8+x^6-x^4+x^2-1}{x^4-1}=\frac{\left(x^{10}-x^8\right)+\left(x^6-x^4\right)+\left(x^2-1\right)}{\left(x^2-1\right)\left(x^2+1\right)}=\frac{\left(x^2-1\right)\left(x^8+x^4+1\right)}{\left(x^2-1\right)\left(x^2+1\right)}=\frac{x^8+x^4+1}{x^2+1}\)
