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

The common denominator for x+5, x^2-25, and 3 is (x+5)(x-5)(3).

Now we can rewrite the equation with the common denominator:

[(3)(x^2-25) + 3(x+5) + 10(x+5)] / [(x+5)(x-5)(3)] = 4/3

Expanding the numerators:

[3x^2 - 75 + 3x + 15 + 10x + 50] / [(x+5)(x-5)(3)] = 4/3

Combining like terms:

(3x^2 + 13x - 10) / [(x+5)(x-5)(3)] = 4/3

Now we can cross multiply to solve for x:

3(3x^2 + 13x - 10) = 4[(x+5)(x-5)(3)]

9x^2 + 39x - 30 = 4(3x^2 - 25)

9x^2 + 39x - 30 = 12x^2 - 100

Rearranging:

3x^2 - 39x - 70 = 0

Now we can solve for x using the quadratic formula:

x = (-(-39) ± √((-39)^2 - 4(3)(-70))) / 2(3)

x = (39 ± √(1521 + 840)) / 6

x = (39 ± √2361) / 6

x = (39 ± 49) / 6

x = 16 or x = -10/3

Therefore, the solutions to the equation are x = 16 and x = -10/3.