To solve this equation, first find a common denominator for the fractions on the left side of the equation.

The common denominator for (x-1)(x+3) and (x-2)(x+4) is (x-1)(x+3)(x-2)(x+4).

Rewrite the equation with the common denominator:

6(x-2)(x+4) - 24(x-1)(x+3) = (x-1)(x+3)(x-2)(x+4)

Expand the terms on the left side of the equation:

6(x^2 + 2x - 4) - 24(x^2 + 3x - 1) = (x-1)(x+3)(x-2)(x+4)

6x^2 + 12x - 24 - 24x^2 - 72x + 24 = (x-1)(x+3)(x-2)(x+4)

Combine like terms:

-18x^2 - 60x = (x-1)(x+3)(x-2)(x+4)

Now expand the right side:

-18x^2 - 60x = (x^2 - 2x + 3x - 3)(x^2 + 4x - 2x - 8)

Simplify the right side further:

-18x^2 - 60x = (x^2 + x - 3)(x^2 + 2x - 8)

Expand the right side once more:

-18x^2 - 60x = x^4 + 2x^3 - 8x^2 + x^3 + 2x^2 - 8x - 3x^2 - 6x + 24

Combine like terms on the right side:

-18x^2 - 60x = x^4 + 3x^3 - 10x^2 - 14x + 24

Now, set the equation equal to zero by subtracting the right side from the left side:

x^4 + 3x^3 - 10x^2 - 14x + 24 + 18x^2 + 60x = 0

Rearrange the terms in descending order:

x^4 + 3x^3 + 8x^2 + 46x + 24 = 0

We have now simplified the equation to the form of a quartic polynomial. The next steps would involve factoring or using numerical methods to find the solutions for x.