1.
\(\dfrac{x^2+2x+1}{5x^3+5x^2}=\dfrac{\left(x+1\right)^2}{5x^2\left(x+1\right)}=\dfrac{x+1}{5x^2}\)
2.
\(\dfrac{x^2-6x+9}{4x^2-12x}=\dfrac{\left(x-3\right)^2}{4x\left(x-3\right)}=\dfrac{x-3}{4x}\)
3.
\(\dfrac{x^2+5x+6}{x^2+4x+4}=\dfrac{\left(x+2\right)\left(x+3\right)}{\left(x+2\right)^2}=\dfrac{x+3}{x+2}\)
4.
\(\dfrac{x^2-6x+9}{x^2-8x+15}=\dfrac{\left(x-3\right)^2}{\left(x-3\right)\left(x-5\right)}=\dfrac{x-3}{x-5}\)
5.
\(\dfrac{3x^2+5x-2}{x^2-3x-10}=\dfrac{\left(3x-1\right)\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}=\dfrac{3x-1}{x-5}\)
6.
\(\dfrac{x^2-8x+12}{x^2-2x-24}=\dfrac{\left(x-2\right)\left(x-6\right)}{\left(x-6\right)\left(x+4\right)}=\dfrac{x-2}{x+4}\)
7.
\(\dfrac{x^3-2x^2+x}{x^2-1}=\dfrac{x\left(x^2-2x+1\right)}{\left(x-1\right)\left(x+1\right)}=\dfrac{x\left(x-1\right)^2}{\left(x-1\right)\left(x+1\right)}=\dfrac{x\left(x-1\right)}{x+1}\)
8.
\(\dfrac{x^3-x^2-x+1}{1-x^3}=\dfrac{x^2\left(x-1\right)-\left(x-1\right)}{-\left(x-1\right)\left(x^2+x+1\right)}=\dfrac{\left(x^2-1\right)\left(x-1\right)}{-\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{1-x^2}{x^2+x+1}\)
9.
\(\dfrac{x^3-x^2-x+1}{1+x^3}=\dfrac{x^2\left(x-1\right)-\left(x-1\right)}{\left(x+1\right)\left(x^2-x+1\right)}=\dfrac{\left(x^2-1\right)\left(x-1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\)
\(=\dfrac{\left(x+1\right)\left(x-1\right)^2}{\left(x+1\right)\left(x^2-x+1\right)}=\dfrac{\left(x-1\right)^2}{x^2-x+1}\)
10.
\(\dfrac{x^3-4x^2+4x}{x^2-4}=\dfrac{x\left(x^2-4x+4\right)}{\left(x-2\right)\left(x+2\right)}=\dfrac{x\left(x-2\right)^2}{\left(x-2\right)\left(x+2\right)}=\dfrac{x\left(x-2\right)}{x+2}\)
11.
\(\dfrac{x^3+3x^2+3x+1}{4x^3+4x^2}=\dfrac{\left(x+1\right)^3}{4x^2\left(x+1\right)}=\dfrac{\left(x+1\right)^2}{4x^2}\)
12.
\(\dfrac{7x^2-7x^3}{x^3-3x^2+3x-1}=\dfrac{-7x^2\left(x-1\right)}{\left(x-1\right)^3}=\dfrac{-7x^2}{\left(x-1\right)^2}\)
