Absolute value is the easiest / hardest concept. The GMAT uses absolute value as a proxy for logical reasoning. Students frequently get fooled by absolute value questions, so my recommendation is whenever possible rewrite the equation without absolute value. This investment of time, in my experience, greatly increases accuracy on these questions.
In this case, if we get rid of absolute value we are left with two equations:
1. x + 2 = 3x + 14
2. x + 2 = -(3x + 14)
Please note that the two other possibilities of -(x + 2) = 3x + 14 and -(x + 2) = -(3x + 14) reduce to either 1 or 2.
Checking in the with the answer choices, only B and C are now alive. Either 1 and 2 have unique values or both solve for the same value. The latter is less likely in real life but could a devious testmaker trick, so you have to check.
1. becomes -2x = 12 then x = -6
2. becomes x + 2 = -3x - 14 then 4x = -16 then x = -4
C is correct
_________________