Let's expand and simplify the left side of the equation:

(x+5)^2 = x^2 + 10x + 25
(x-2)^2 = x^2 - 4x + 4
(x-7)(x+7) = x^2 - 49

Adding them together:

x^2 + 10x + 25 + x^2 - 4x + 4 + x^2 - 49 = 11x + 80

Combining like terms:

3x^2 + 6x - 20 = 11x + 80

Now let's move all terms to one side to set the equation to zero:

3x^2 + 6x - 20 - 11x - 80 = 0
3x^2 - 5x - 100 = 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:

x = (5 ± √((-5)^2 - 4(3)(-100))) / 2(3)
x = (5 ± √(25 + 1200)) / 6
x = (5 ± √1225) / 6
x = (5 ± 35) / 6

There are two possible solutions:

x = (5 + 35) / 6 = 40 / 6 = 20/3 or x = 6.67
x = (5 - 35) / 6 = -30 / 6 = -5

Therefore, the solutions for the equation are x = 20/3 or x = -5.