ОДЗ: \(x + 1 > 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x > -1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {-1;\infty } \right).\)
\({2^{{{\log }_4}\left( {x + 1} \right)}} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^{{{\log }_{{2^2}}}\left( {x + 1} \right)}} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^{\frac{1}{2}{{\log }_2}\left( {x + 1} \right)}} = 3\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{2^{{{\log }_2}\sqrt {x + 1} }} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\sqrt {x + 1} = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x + 1 = 9\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x = 8.\)
Ответ: 8.