a) \(\left(x^3+12x^2-5x\right):x\)
\(=\dfrac{x\left(x^2+12x-5\right)}{x}\)
\(=x^2+12x-5\)
b) \(\left(3x^4y^3-9x^2y^2+25xy^3\right):xy^2\)
\(=\dfrac{xy^2\cdot\left(3x^3y-9x+25y\right)}{xy^2}\)
\(=3x^3y+9x+25y\)
c) \(\left(5x^5y^4z+\dfrac{1}{2}x^4y^2z^3-2xy^3z\right):\dfrac{1}{2}xy^2z\)
\(=\dfrac{\dfrac{1}{2}xy^2z\left(10x^4y^2+x^3z^2-4y\right)}{\dfrac{1}{2}xy^2z}\)
\(=10x^4y^2+x^3z^2-4y\)
d) \(\left[\left(x-y\right)^3+3\left(x-y\right)\right]:\dfrac{1}{3}\left(x-y\right)\)
\(=\dfrac{\dfrac{1}{3}\left(x-y\right)\left[3\left(x-y\right)^2+9\right]}{\dfrac{1}{3}\left(x-y\right)}\)
\(=3\left(x-y\right)^2+9\)
e) \(\left(8x^3-27y^3\right):\left(2x-3y\right)\)
\(=\dfrac{\left(2x-3y\right)\left(4x^2+6xy+9y^2\right)}{2x-3y}\)
\(=4x^2+6xy+9y^2\)
f) \(\left[5\left(x+2y\right)^6-6\left(x+2y\right)^5\right]:2\left(x+2y\right)^4\)
\(=\dfrac{2\left(x+2y\right)^4\cdot\left[\dfrac{5}{2}\left(x+2y\right)^2-3\left(x+2y\right)\right]}{2\left(x+2y\right)^4}\)
\(=\dfrac{5}{2}\left(x+2y\right)^2-3\left(x+2y\right)\)
