1. A = 2x / (x² - 3x) + 2x / (x² - 4x + 3) + x / (x - 1)

A = 2x / (x(x - 3)) + 2x / ((x - 1)(x - 3)) + x / (x - 1)
A = 2 / (x - 3) + 2x / ((x - 1)(x - 3)) + x(x - 3) / ((x - 1)(x - 3))
A = (2(x - 1) + 2x + x(x - 3)) / ((x - 1)(x - 3))
A = (2x - 2 + 2x + x² - 3x) / ((x - 1)(x - 3))
A = (x² + x - 2) / ((x - 1)(x - 3))
A = (x - 1)(x + 2) / ((x - 1)(x - 3))
A = (x + 2) / (x - 3)

2. B = x / (x + 2) + 2 / (x - 2) - 4x / (4 - x²)

B = x / (x + 2) + 2 / (x - 2) + 4x / (x² - 4)
B = x / (x + 2) + 2 / (x - 2) + 4x / ((x - 2)(x + 2))
B = x(x - 2) / ((x + 2)(x - 2)) + 2(x + 2) / ((x + 2)(x - 2)) + 4x / ((x - 2)(x + 2))
B = (x² - 2x + 2x + 4 + 4x) / ((x + 2)(x - 2))
B = (x² + 4x + 4) / ((x + 2)(x - 2))
B = (x + 2)² / ((x + 2)(x - 2))
B = (x + 2) / (x - 2)

3. C = (1 + x) / (3 - x) + (1 - 2x) / (3 + x) - x(1 - x) / (9 - x²)

C = (1 + x) / (3 - x) + (1 - 2x) / (3 + x) - x(1 - x) / ((3 - x)(3 + x))
C = (1 + x)(3 + x) / ((3 - x)(3 + x)) + (1 - 2x)(3 - x) / ((3 - x)(3 + x)) - x(1 - x) / ((3 - x)(3 + x))
C = (3 + x + 3x + x² + 3 - x - 6x + 2x² - x + x²) / ((3 - x)(3 + x))
C = (4x² - 3x + 6) / ((3 - x)(3 + x))

4. D = 5 / (2x² + 6x) - (4 - 3x²) / (x² - 9) - 3 / (x + 3)

D = 5 / (2x(x + 3)) - (4 - 3x²) / ((x - 3)(x + 3)) - 3 / (x + 3)
D = 5(x - 3) / (2x(x - 3)(x + 3)) - 2x(4 - 3x²) / (2x(x - 3)(x + 3)) - 6x(x - 3) / (2x(x - 3)(x + 3))
D = (5x - 15 - 8x + 6x³ - 6x² + 18x) / (2x(x - 3)(x + 3))
D = (6x³ - 6x² + 15x - 15) / (2x(x - 3)(x + 3))

5. E = (3x + 2) / (x² - 2x + 1) - 6 / (x² - 1) - (3x - 2) / (x² + 2x + 1)

E = (3x + 2) / (x - 1)² - 6 / ((x - 1)(x + 1)) - (3x - 2) / (x + 1)²
E = (3x + 2)(x + 1)² / ((x - 1)²(x + 1)²) - 6(x - 1)(x + 1) / ((x - 1)²(x + 1)²) - (3x - 2)(x - 1)² / ((x - 1)²(x + 1)²)
E = (3x³ + 8x² + 7x + 2 - 6x² + 6 - 3x³ + 8x² - 7x + 2) / ((x - 1)²(x + 1)²)
E = (10x² + 10) / ((x - 1)²(x + 1)²)
E = 10(x² + 1) / ((x - 1)²(x + 1)²)

6. F = (x² + x - 2) / (x² - x - 6) * (x² + 3x - 4) / (x² + x - 12) * (y² - 2y - 15) / (y² - 4) * (y² + y - 6)

F = ((x + 2)(x - 1)) / ((x - 3)(x + 2)) * ((x + 4)(x - 1)) / ((x + 4)(x - 3)) * ((y - 5)(y + 3)) / ((y - 2)(y + 2)) * ((y + 3)(y - 2)) / ((y + 3)(y - 2))
F = 1

7. G = x² + (x⁴ + 1) / (1 - x²)

G = x² + (x⁴ + 1) / (1 - x²)**

G = (x²(1 - x²) + x⁴ + 1) / (1 - x²)

G = (x² - x⁴ + x⁴ + 1) / (1 - x²)

G = (x² + 1) / (1 - x²)

8. H = 4x / (x² + 2x) + 3 / (2 - x) + 12x / (x³ - 4x)

H = 4x / (x(x + 2)) + 3 / (2 - x) + 12x / (x(x - 2)(x + 2))
H = 4 / (x + 2) - 3 / (x - 2) + 12 / ((x - 2)(x + 2))
H = 4(x - 2) / ((x - 2)(x + 2)) - 3(x + 2) / ((x - 2)(x + 2)) + 12 / ((x - 2)(x + 2))
H = (4x - 8 - 3x - 6 + 12) / ((x - 2)(x + 2))
H = (x - 2) / ((x - 2)(x + 2))
H = 1 / (x + 2)

9. I = (4x² - 3x + 17) / (x³ - 1) + (2x - 1) / (x² + x + 1) + 6x / (x - x²)

I = (4x² - 3x + 17) / ((x - 1)(x² + x + 1)) + (2x - 1) / (x² + x + 1) - 6 / (x - 1)
I = (4x² - 3x + 17 + (2x - 1)(x - 1) - 6(x² + x + 1)) / ((x - 1)(x² + x + 1))
I = (4x² - 3x + 17 + 2x² - 3x + 1 - 6x² - 6x - 6) / ((x - 1)(x² + x + 1))
I = (-12x + 12) / ((x - 1)(x² + x + 1))
I = -12 / (x² + x + 1)

10. K = (x + 9y) / (x² - 9y²) - 3y / (x² + 3xy)

K = (x + 9y) / ((x - 3y)(x + 3y)) - 3y / (x(x + 3y))
K = x / (x(x - 3y)) - 3y / (x(x - 3y))
K = 1 / (x - 3y)