\[Уравнение\ имеет\ \]
\[единственный\ корень\ \]
\[при\ D = 0.\]
\[(m + 4)x^{2} - (m + 5)x + 1 = 0\]
\[m + 4 = 0 \Longrightarrow m = - 4.\]
\[D = (m + 5)^{2} - 4 \cdot (m + 4) \cdot 1 =\]
\[= m^{2} + 10m + 25 - 4m - 16 =\]
\[= m^{2} + 6m + 9 = (m + 3)^{2}\]
\[(m + 3)^{2} = 0\]
\[m + 3 = 0\]
\[m = - 3.\]
\[Ответ:\ при\ m = - 3;\ m = - 4.\]