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To solve this equation, we first need to find a common denominator for the fractions on both sides of the equation.

Given equation: 12/(x^2 + x - 10) - 6/(x^2 + x - 7) = 5/(x^2 + x - 11)

The denominators are (x + 5)(x - 2), (x + 7)(x - 1), and (x + 5)(x - 6), respectively.

Multiplying each term by the common denominator (x + 5)(x - 2)(x + 7)(x - 1)(x + 5)(x - 6), we get:

12(x + 7)(x - 1) - 6(x + 5)(x - 6) = 5(x + 5)(x - 2)

Expanding and simplifying the equation, we get:

12(x^2 + 6x - 7) - 6(x^2 - x - 30) = 5(x^2 + 3x - 10)

12x^2 + 72x - 84 - 6x^2 + 6x + 180 = 5x^2 + 15x - 50

Combining like terms, we get:

6x^2 + 78x + 96 = 5x^2 + 15x - 50

Rearranging the terms, we get:

6x^2 - 5x^2 + 78x - 15x + 96 + 50 = 0

x^2 + 63x + 146 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values of a, b, and c, we get:

x = (-63 ± √(63^2 - 4(1)(146))) / 2(1)

x = (-63 ± √(3969 - 584)) / 2

x = (-63 ± √3385) / 2

Therefore, the solutions to the equation are:

x = (-63 + √3385) / 2 and x = (-63 - √3385) / 2