2011 AMC 10B Problem 19

Because $x^2=|x|^2,$ squaring the equation gives $$\begin{aligned} 5|x|+8 &= |x|^2-16,\\ |x|^2-5|x|-24&=0,\\ (|x|-8)(|x|+3)&=0. \end{aligned}$$ The algebraic possibilities for $|x|$ are $8$ and $-3,$ but an absolute value cannot be negative. Hence $|x|=8,$ so the roots are $8$ and $-8$, with product $-64.$ Thus, the correct answer is A .