\({2^{{x^2} + 3}}-{8^{x + 1}} = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{2^{{x^2} + 3}} = {2^{3x + 3}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{x^2} + 3 = 3x + 3\,\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\,\,{x^2}-3x = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,x\left( {x-3} \right) = 0\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 0,\,\,\,\,\,\,\,}\\{x-3 = 0}\end{array}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 0,}\\{x = 3.}\end{array}} \right.} \right.\)
Суммы корней: \(0 + 3 = 3.\)
Ответ: 3.