To solve this equation, we first need to find a common denominator for all three fractions:

(x-2)(x+5) for the left side of the equation, and
(x-2)(x+5) for the right side of the equation.

After finding the common denominator, the equation becomes:

(x(x+5) - 8(x-2)) / (x-2)(x+5) = 14 / (x+5)(x-2)

Expanding and simplifying the numerators of the fractions on the left side, the equation becomes:

(x^2 + 5x - 8x + 16) / (x-2)(x+5) = 14 / (x+5)(x-2)

(x^2 - 3x + 16) / (x-2)(x+5) = 14 / (x+5)(x-2)

Now we can simplify the equation by clearing the denominators:

(x^2 - 3x + 16) = 14

Rearranging the equation gives:

x^2 - 3x + 16 - 14 = 0

x^2 - 3x + 2 = 0

Now, we factor the quadratic equation:

(x-2)(x-1) = 0

This gives us two possible solutions:

x = 2 or x = 1

Therefore, the solutions to the equation are x = 2 and x = 1.