The technique described in the link does not apply to this problem because the technique is about solving inequalities. This problem requires no use of inequalities.
I also don't really understand the technique you mentioned where you checked $10|x+4|$ for positive and negative values. I know what that means but I don't know how it helps.
The best way (that I can think of) to approach this is to simply solve the equation and add up the valid solutions.
First let $y=|x+4|$ and subtract 24 from both sides. Then the equation becomes $y^2 - 10y -24=0.$ Solve by factoring (or however) to get $y=-2$ and $y=12.$
Back-substitute to solve for $x$. The two equations you'll get are $|x+4|=-2$ and $|x+4|=12.$ The first one has no solutions because the LHS is an absolute value and can therefore never be equal to a negative number. The second one yields solutions $x=8$ and $x=-16.$ Verify both solutions in the original equation from the beginning of the problem to make sure we don't have extraneous solutions. We don't. Therefore the solutions are 8 and -16, and so the sum is -8.
Some background info on critical points, etc. You asked about what this method should be called. As with a few things in precalculus, there's really no standard name for this type of technique. I've heard critical points, critical values, critical numbers, key values, and now transitional points. Different books and instructors say different things but AFAIK, high-level research rarely (if ever) requires in-depth discussion of topics like this, which may be why there's no standard name. Personally I find it unfortunate that "critical points" (or "critical" anything) is used in this context because the terms critical point and critical value actually do have a standard definition in calculus, and it's very different from what's happening here in this problem. This calculus definition is what Moo was getting at with his comment to your original question but, as I stated in my follow-up comment, the fact that it gets the correct answer is a coincidence here.