Let's substitute (x+1)^2 as a variable. Let y = (x+1)^2

The equation becomes:

y^2 - 5y - 36 = 0

Now we can solve this quadratic equation for y by factoring:

(y-9)(y+4) = 0

So, y = 9 or y = -4

Now substitute back y = (x+1)^2:

(x+1)^2 = 9
Taking square root:
x+1 = ±3
x = 3 - 1 or x = -3 - 1
x = 2 or x = -4

or

(x+1)^2 = -4
Taking square root:
x+1 = ±2i
x = -1 ± 2i

Therefore, the solutions to the original equation (x+1)^4 - 5(x+1)^2 - 36 = 0 are:

x = 2, x = -4, x = -1 + 2i, x = -1 - 2i