Answer
$x=-5$ or $ x=-1$ or $ x=-9$
Work Step by Step
$(x+5)^4-16(x+5)^2=0$
Factor $(x+5)^2$
$\Rightarrow (x+5)^2\bigg((x+5)^2-16\bigg)=0$
Expand $\Rightarrow (x+5)^2\bigg((x^2+10x+25)-16\bigg)=0$
Collect $\Rightarrow (x+5)^2\bigg(x^2+10x+9\bigg)=0$
Factor $\Rightarrow (x+5)^2\bigg((x+1)(x+9)\bigg)=0$
$\Rightarrow (x+5)^2(x+1)(x+9)=0$
Zero-Product Property $ x+5=0$ or $ x+1=0$ or $ x+9=0$
Solve $x=-5$ or $ x=-1$ or $ x=-9$