Let's solve the equations step by step:

1) (x+5)(x-4) + x^2 + 18 = 0
Expand the left side:
x^2 - 4x + 5x - 20 + x^2 + 18 = 0
2x^2 + x - 2 = 0
Factor the quadratic equation:
(x + 2)(2x - 1) = 0
From here, we find two possible solutions:
x = -2 or x = 1/2

2) (x+7)(x-7) + (x-3)^2 = 53
Expand the left side:
x^2 - 7x + 7x - 49 + x^2 - 6x + 9 = 53
2x^2 - 6x - 40 = 0
Divide by 2:
x^2 - 3x - 20 = 0
Factor the quadratic equation:
(x - 5)(x + 4) = 0
From here, we find two possible solutions:
x = 5 or x = -4

Therefore, the solutions to the system of equations are:
x = -2, 1/2, 5, -4