(x+4)^2 > 4

Make it a equality first.

(x+4)^2 = 4

Solve it we get

(X+2)(X+6) = 0

This means

X = -2 or X = -6.

Now this -2 and -6 are our check points. what we have done is that we have broken the number line into 3 intervals.

(-infiniy,-6) (-6,-2) and (-2,+infinity). we just need to check that within which interval this expression is positive (why positive ? because the initia inequality asked us to find > 4. Had the initial inequaiy asked us to find <4 then we would have find out within which of these 3 interval we have a -ve value).

Now make a grid as follows or use the check point method. I do not recommend check point method so lets go with the grid.

refer my attachment as I do not know how to put a picture here.