a: \(\dfrac{2x^2+2x-4}{x+2}\)
\(=\dfrac{2x^2+4x-2x-4}{x+2}\)
\(=\dfrac{2x\left(x+2\right)-2\left(x+2\right)}{x+2}\)
\(=2x-2\)
b: \(\dfrac{2x^3-5x^2-x+1}{2x+1}\)
\(=\dfrac{2x^3+x^2-6x^2-3x+2x+1}{2x+1}\)
\(=\dfrac{x^2\left(2x+1\right)-3x\left(2x+1\right)+\left(2x+1\right)}{2x+1}\)
\(=x^2-3x+1\)
c: \(\dfrac{x^3-2x+4}{x+2}\)
\(=\dfrac{x^3+2x^2-2x^2-4x+2x+4}{x+2}\)
\(=\dfrac{x^2\left(x+2\right)-2x\left(x+2\right)+2\left(x+2\right)}{x+2}\)
\(=x^2-2x+2\)
d: \(\dfrac{x^3-3x^2}{x-3}\)
\(=\dfrac{x^2\left(x-3\right)}{x-3}\)
\(=x^2\)
e: \(\dfrac{x^4-x-14}{x-2}\)
\(=\dfrac{x^4-2x^3+2x^3-4x^2+4x^2-8x+7x-14}{x-2}\)
\(=\dfrac{x^3\left(x-2\right)+2x^2\left(x-2\right)+4x\left(x-2\right)+7\left(x-2\right)}{x-2}\)
\(=x^3+2x^2+4x+7\)
f: \(\dfrac{x^3+x^2-12}{x-2}\)
\(=\dfrac{x^3-2x^2+3x^2-6x+6x-12}{x-2}\)
\(=\dfrac{x^2\left(x-2\right)+3x\left(x-2\right)+6\left(x-2\right)}{x-2}\)
\(=x^2+3x+6\)
g: \(\dfrac{-6x^4+5x^3+17x^2-23x+7}{-3x^2-2x+7}\)
\(=\dfrac{6x^4-5x^3-17x^2+23x-7}{3x^2+2x-7}\)
\(=\dfrac{6x^4+4x^3-14x^2-9x^3-6x^2+21x+3x^2+2x-7}{3x^2+2x-7}\)
\(=\dfrac{2x^2\left(3x^2+2x-7\right)-3x\left(3x^2+2x-7\right)+\left(3x^2+2x-7\right)}{3x^2+2x-7}\)
\(=2x^2-3x+1\)
