\(\begin{array}{l}
{4^x} + {4^{ - x}} = 14 \Leftrightarrow {2^{2x}} + {2^{ - 2x}} + 2 = 16\\
\Leftrightarrow {\left( {{2^x} + {2^{ - x}}} \right)^2} = 16 \Rightarrow {2^x} + {2^{ - x}} = 4\\
P = \frac{{10 - \left( {{2^x} + {2^{ - x}}} \right)}}{{3 + {2^x} + {2^{ - x}}}} = \frac{{10 - 4}}{{3 + 4}} = \frac{6}{7}
\end{array}\)