Answer
$(-\infty,-5]\cup[1,\infty)$
Work Step by Step
Step 1. From $-4|x+2|+5\leq-7$, we have $-4|x+2|\leq-12$, $|x+2|\geq3$ which gives $x+2\geq3$ or $x+2\leq-3$. Thus $x\leq-5\ \cup\ x\geq1$, and in interval notation $(-\infty,-5]\cup[1,\infty)$
Step 2. See graph for the interval on a number line.