\(\begin{array}{*{20}{l}}
\begin{array}{l}
\log _2^2x - 2{\log _4}(4x) - 4 = 0\\
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{x > 0}\\
{{{\left( {{{\log }_2}x} \right)}^2} - (2 - {{\log }_2}x) - 4 = 0}
\end{array}} \right.
\end{array}\\
{ \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{x > 0}\\
{{{({{\log }_2}x)}^2} - {{\log }_2}x - 6 = 0}
\end{array}} \right.}
\end{array}\)

\(\begin{array}{l}
 \Leftrightarrow \left\{ {\begin{array}{*{20}{l}}
{x > 0}\\
{({{\log }_2}x - 3)({{\log }_2}x + 2) = 0}
\end{array}} \right.\\
 \Leftrightarrow \left[ {\begin{array}{*{20}{c}}
{x = 8}\\
{x = \frac{1}{4}}
\end{array}} \right. \Rightarrow {x_1}.{x_2} = 8.\frac{1}{4} = 2
\end{array}\)