To solve this equation, we need to find a common denominator for all the fractions involved. In this case, the common denominator is (x-4)(x-2)(6).

Multiplying each term by the common denominator, we get:

(14/3)(x-4)(x-2)(6) - (12)(x-4)(x-2)(6) - (2+x)(x-4)(x-2)(6)/(x-4)(x-2)(6) = (3)(x-4)(x-2)(6) - (5/6)(x-4)(x-2)(6)

Simplifying both sides of the equation, we have:

14(x-4)(x-2) - 12(x-4)(x-2) - (2+x)(x-4)(x-2) = 3(x-4)(x-2) - 5(x-4)(x-2)

Expanding and combining like terms, we get:

14x^2 - 56x + 56 - 12x^2 + 48x - 48 - 2x^3 + 8x^2 - 8x - 2x^2 + 8x - 8 = 3x^2 - 12x - 8x^2 + 32x - 20x + 80

Simplifying further, we have:

-2x^3 + 8x^2 - 2x^2 + 3x^2 + 14x^2 - 12x^2 - 56x + 48x + 8x - 8x + 32x - 20x + 56 - 48 - 8 - 80 = 0

Combining like terms, we get:

-2x^3 + 11x^2 + 34x - 80 = 0

This is a cubic equation. To solve it, we can use factoring, synthetic division, or numerical methods such as Newton's method.