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.
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.