(x-4)/(x-2) + 8/x = 2/(x+2)

Multiply each term in the equation by x(x-2)(x+2) to clear the denominators:

x(x-4) + 8(x-2)(x+2) = 2x(x-2)

Expand and simplify:

x^2 - 4x + 8(x^2 - 4) = 2x(x^2 - 2x)

x^2 - 4x + 8x^2 - 32 = 2x^3 - 4x^2

9x^2 - 4x - 32 = 2x^3 - 4x^2

Rearrange and simplify:

2x^3 - 9x^2 + 4x - 32 = 0

(x-1)/(x+4) + (x-9)/(4-x) = 40/(x^2 - 16)

To simplify this equation, first combine the fractions on the left side:

[(x-1)(4-x) + (x-9)(x+4)] / [(x+4)(4-x)] = 40/(x^2 - 16)

[(x^2 - 5x + 4 + x^2 - 5x - 36)] / [(x+4)(4-x)] = 40/(x^2 - 16)

[2x^2 - 10x - 32] / [(x+4)(x-4)] = 40/(x^2 - 16)

Simplify and cross multiply:

(2x^2 - 10x - 32)(x^2 - 16) = 40[(x+4)(x-4)]

Expand and solve:

2x^4 - 10x^3 - 32x^2 + 160x^2 - 800 - 128 = 40(x^2 - 16)

2x^4 - 10x^3 + 128x^2 - 800x - 128 = 40x^2 - 640

Rearrange and simplify:

2x^4 - 10x^3 + 88x^2 - 800x - 640 = 0

This is the simplified form of the equation.