To solve these equations, we can use the quadratic formula. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

For the first equation, X^2 - 4X - 5 = 0, the coefficients are a = 1, b = -4, and c = -5. Plugging these values into the quadratic formula, we get:

X = (4 ± √((-4)² - 41(-5))) / 2*1
X = (4 ± √(16 + 20)) / 2
X = (4 ± √36) / 2
X = (4 ± 6) / 2

Therefore, the solutions for the first equation are X = 5 or X = -1.

For the second equation, X^2 + 7X + 12 = 0, the coefficients are a = 1, b = 7, and c = 12. Plugging these values into the quadratic formula, we get:

X = (-7 ± √(7² - 4112)) / 2*1
X = (-7 ± √(49 - 48)) / 2
X = (-7 ± √1) / 2
X = (-7 ± 1) / 2

Therefore, the solutions for the second equation are X = -6 or X = -1.