\(\left|x+5\right|+\left|x+17\right|=\left|-5-x\right|+\left|x+17\right|\ge\left|-5-x+x+17\right|=12\Rightarrow A_{min}=12\Leftrightarrow\left(-5-x\right)\left(x+17\right)\ge0\Leftrightarrow\left(x+5\right)\left(x+17\right)\le0\Leftrightarrow-17\le x\le-5\)