What number should be added to both sides of the equation to complete the square? x2 + 8x = 4

The number that should be added to both sides is 16 because its half of b, which is then squared. You then get (x+4)^2=20, which you then square root both sides and get x+4=+-20 and then subtract the 4 to get x=-4+-20

In a "complete the square" problem when we have an ax^2 + bx +c situation we can complete the square by adding (b/2)^2

As this equation is in the form:
1x^2 + 8x = 4

We can complete the square by adding (b/2)^2 to both sides.

b = 8, so (b/2)^2 = (8/2)^2 = 16

The (b/2)^2 rule is based on geometry and can be looked up if you search for a tutorial on "completing the square". It's quite tricky.