Isolate the Bars First

In $\left|x+3\right|-2 = 4$, the absolute value is not alone on its side. Before you split, get the absolute value by itself. Add 2 to both sides:

$$\left|x+3\right| = 6$$

Now split into two cases: $x+3 = 6$ or $x+3 = -6$. Subtract 3 in each case to get $x = 3$ or $x = -9$.

The -2 in $\left|x+3\right|-2 = 4$ sits outside the absolute value bars, so it is not part of either case. Add 2 to both sides first to get $\left|x+3\right| = 6$. Then solve $x+3 = 6$ and $x+3 = -6$ separately.