We can start by solving the absolute value equation |x+2|.

When x+2 is greater or equal to 0:
x+2 = x+2

When x+2 is less than 0:
x+2 = -(x+2)

Now let's solve these two cases:

Case 1: x+2 >= 0
x+2 = x+2
x = x

Case 2: x+2 < 0
x+2 = -(x+2)
x+2 = -x-2
2x = -4
x = -2

Now we need to check these solutions in the original equation:

For x = x:
x^2 + 4x = 4 + 2|x+2|
x^2 + 4x = 4 + 2|0|
x^2 + 4x = 4
x^2 + 4x - 4 = 0

Now, let's check the other solution:

For x = -2:
x^2 + 4x = 4 + 2|-2+2|
(-2)^2 + 4(-2) = 4 + 2|0|
4 - 8 = 4
-4 = 4(false)

Therefore, the only solution to the equation x^2 + 4x = 4 + 2|x+2| is x = 0.