a: \(\frac{2x^3+x^2-2x-1}{x^3+2x^2-x-2}\)
\(=\frac{x^2\left(2x+1\right)-\left(2x+1\right)}{x^2\left(x+2\right)-\left(x+2\right)}\)
\(=\frac{\left(2x+1\right)\left(x^2-1\right)}{\left(x+2\right)\left(x^2-1\right)}=\frac{2x+1}{x+2}\)
b: \(\frac{50x-20x^2+2x^3}{x^2-25}=\frac{2x\left(x^2-10x+25\right)}{\left(x-5\right)\left(x+5\right)}\)
\(=\frac{2x\left(x-5\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)}=\frac{2x\left(x-5\right)}{x+5}\)
c: \(\frac{25-\left(x+2\right)^2}{x^2-6x+9}\)
\(=\frac{\left(5-x-2\right)\left(5+x+2\right)}{\left(x-3\right)^2}=\frac{\left(3-x\right)\left(7+x\right)}{\left(3-x\right)^2}=\frac{7+x}{3-x}\)
d: \(\frac{y^3+y^2+4y+4}{y^2-2y-3}\)
\(=\frac{y^2\left(y+1\right)+4\left(y+1\right)}{\left(y-3\right)\left(y+1\right)}=\frac{\left(y^2+4\right)\left(y+1\right)}{\left(y-3\right)\left(y+1\right)}\)
\(=\frac{y^2+4}{y-3}\)
e: \(\frac{4-4x^2-9y^2-12xy}{2x+3y+2}\)
\(=\frac{4-\left(2x+3y\right)^2}{2x+3y+2}=\frac{\left(2+2x+3y\right)\left(2-2x-3y\right)}{2x+3y+2}=2-2x-3y\)
f: \(\frac{x^7-x^4+x^3-1}{x^6+x^5+x^4+x^2+x+1}\)
\(=\frac{x^4\left(x^3-1\right)+\left(x^3-1\right)}{\left.\left(x^2+x+1\right)\right.\left(x^4+1\right)}\)
\(=\frac{\left(x^3-1\right)\left(x^4+1\right)}{\left(x^2+x+1\right)\left(x^4+1\right)}=\frac{x^3-1}{x^2+x+1}\)
=x-1
