\({2^{x + 2}} + {2^x} > 20\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^x} \cdot {2^2} + {2^x} > 20\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^x} \cdot \left( {4 + 1} \right) > 20\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,5 \cdot {2^x} > 20\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^x} > 4\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^x} > {2^2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x > 2\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\, \in \,\left( {2;\infty } \right).\)
Следовательно, наименьшее целое решение \(x = 3.\)
Ответ: 3.