Answer
Solutions are $1,-9$
Work Step by Step
$|x+4|-4 = 1$
For isolating absolute value add $4$ to both sides.
$|x+4|-4 +4= 1+4$
$|x+4| = 5$
$x+4 = 5$ or $x+4 = -5$
$x = 5-4$ or $x = -5-4$
$x=1$ or $x=-9$
Solutions are $1,-9$