a: \(A=\left(x-5\right)\left(2x+3\right)-2x\left(-x-3\right)+\dfrac{x^3+4x^2-19x+14}{x^2-3x+2}\)
\(=2x^2+3x-10x-15+2x^2+6x+\dfrac{x^3-3x^2+2x+7x^2-21x+14}{x^2-3x+2}\)
\(=4x^2-x-15+\dfrac{\left(x^2-3x+2\right)\left(x+7\right)}{x^2-3x+2}\)
\(=4x^2-x-15+x+7=4x^2-8\)
b: \(A=\left(3x-5\right)\left(2x+11\right)-\left(2x+3\right)\left(-3x+7\right)-\dfrac{5x^5+7x^2-6x^3}{x^2}\)
\(=6x^2+33x-10x-55-\left(-6x^2+14x-9x+21\right)-\dfrac{x^2\left(5x^3+7-6x\right)}{x^2}\)
\(=6x^2+23x-55+6x^2-5x-21-5x^3-7+6x\)
\(=-5x^3+12x^2+24x-83\)
c: \(A=\left(3-x\right)\left(2x-1\right)\left(3x+2\right)-6x\left(x-1\right)-\dfrac{x^3-27}{x^2+3x+9}\)
\(=\left(3-x\right)\left(6x^2+4x-3x-2\right)-6x^2+6x-\dfrac{\left(x-3\right)\left(x^2+3x+9\right)}{x^2+3x+9}\)
\(=\left(-x+3\right)\left(6x^2+x-2\right)-6x^2+6x-x+3\)
\(=\left(-x+3\right)\left(6x^2+x-2\right)-6x^2+5x+3\)
\(=-6x^3-x^2+2x+18x^2+3x-6-6x^2+5x+3\)
\(=-6x^3+11x+10x-3\)
d: \(A=\dfrac{x^3+7x^2+7x-6}{x+2}-4\left(x-4\right)\left(x+4\right)-x\left(2x-1\right)\)
\(=\dfrac{x^3+2x^2+5x^2+10x-3x-6}{x+2}-4\left(x^2-16\right)-2x^2+x\)
\(=\dfrac{\left(x+2\right)\left(x^2+5x-3\right)}{x+2}-4x^2+64-2x^2+x\)
\(=x^2+5x-3-6x^2+x+64\)
\(=-5x^2+6x+61\)
e: \(A=-2x\left(x-7\right)\left(x+3\right)-\left(5x-1\right)\left(x+4\right)-3x\left(x-27\right)\)
\(=-2x\left(x^2+3x-7x-21\right)-\left(5x^2+20x-x-4\right)-3x^2+81x\)
\(=-2x\left(x^2-4x-21\right)-5x^2-19x+4-3x^2+81x\)
\(=-2x\left(x^2-4x-21\right)-8x^2+62x+4\)
\(=-2x^3+8x^2+42x-8x^2+62x+4\)
\(=-2x^3+104x+4\)
