(-3x + 4)(x - 2) - x(x² - 2)(-4x + 1) + (2x - 5)(x² - 4x)

= -3x² + 10x - 8 - x(-4x³ + x² + 8x - 2) + 2x³ - 13x² + 20x

= 2x³ - 16x² + 30x - 8 + 4x⁴ - x³ - 8x² + 2x

= 4x⁴ + x³ - 24x² + 28x - 8

-7x(2x²- 4x - 5) - (x - 5)(-2x + 3) + (3x - 2)(x+4) - 4x² (x - 3)

=-14x³+28x²+35x+2x²-3x-10x+15+3x²+12x-2x-8-4x³+12x²

=-18x³+45x²+32x+7

(x - 6)(3x - 1) - 2x²(x - 2) + x(x - 1)(2x + 3) - 5x(x² - 4x + 1) 

= (3x² - 19x + 6) - (2x³ - 4x²) + (x² - x)(2x + 3) - (5x³ - 20x² + 5x) 

= 3x² - 19x + 6 - 2x³ + 4x² + 2x³ + 3x² - 2x² - 3x - 5x³ + 20x² + 5x

= -7x³ + 28x²+ 17x + 6 

(3x - 7)(x + 3) + 5x(x² - 2x - 4) - (4x - 5)(x-4)-(x + 2)(x + 1)

=3x²+9x-7x-21+5x³-10x²-20x-4x²+16x+5x-20-x²-x-2x-2

=5x³-13x²-43

(-2x² + 5)(x - 6) - 4x(x² - 7x + 2) - 2x(x  + 3)(x - 2) - (x³ - x²)

= -2x³ + 12x² + 5x - 30 - 4x³ + 28x² - 8x - 2x(x² +  x - 6) - x³ + x²

= -8x³ + 41x² - 3x - 30 - 2x³ - 2x² + 12x

= -10x³ + 39x² + 9x - 30

5x² (x - 7) - 4x(x - 2)(x + 3) - (x - 5)(x + 6) + (2x-1)(x + 5)

=5x³-35x²-4x(x²+3x-2x-6)-(x²+6x-5x-30)+2x²+10x-x-5

=5x³-35x²-4x³-12x²+8x²+24x-x²-6x+5x+30+2x²+9x-5

=x³+32x²+32x+25

a) (2x - 1)(3x + 5) - 2(-4x + 1)² = 6x² + 10x - 3x - 5 - 2(16x² - 8x + 1) = 6x² - 3x - 5 - 32x² + 16x - 2 = -26x² + 13x - 7

b) \(\frac{x^2-16}{4x-x^2}=\frac{\left(x-4\right)\left(x+4\right)}{-x\left(x-4\right)}=-\frac{x+4}{x}\)

c) \(\frac{2x-9}{x^2-5x+6}+\frac{2x+1}{x-3}+\frac{x+3}{2-x}\)

= \(\frac{2x-9}{x^2-2x-3x+6}+\frac{\left(2x+1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}-\frac{\left(x+3\right)\left(x-3\right)}{\left(x-3\right)\left(x-2\right)}\)

= \(\frac{2x-9+2x^2-3x-2-x^2+9}{\left(x-3\right)\left(x-2\right)}\)

= \(\frac{x^2-x-2}{\left(x-3\right)\left(x-2\right)}\)

= \(\frac{x^2-2x+x-2}{\left(x-3\right)\left(x-2\right)}\)

= \(\frac{\left(x+1\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}=\frac{x+1}{x-3}\)

d) (x - 1)³ - (x + 1)³ + 6(x + 1)(x - 1)

= (x - 1 - x - 1)[(x - 1)² + (x - 1)(x + 1) + (x + 1)²] + 6(x² - 1)

= -2(x² - 2x + 1  + x² - 1 + x² + 2x + 1) + 6x² - 6

= -2(3x² + 1) + 6x² - 6

= -6x² - 2 + 6x²  - 6

= -8

e) (2x + 7)² - (4x + 14)(2x - 8) + (8 - 2x)²

= (2x + 7)² - 2(2x + 7)(2x - 8) + (2x - 8)²

= (2x + 7 - 2x + 8)²

= 15² = 225

(5x - 2)(x + 1) - 3x(x² - x - 3) - 2x(x - 5)(x + 4) + 5x(x² - 7)

= 5x² + 3x - 2 - 3x³ + 3x² + 9x - 2x(x² - x - 20) + 5x³ - 35x

= 2x³ + 8x² - 23x - 2 - 2x³ + 2x² + 40x = 10x² + 17x - 2