\[\boxed{\text{349\ (349).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[1)\ x^{3} - x + 3 = 0 \Longrightarrow среди\ \]
\[делителей\ 3\ нет\ целых\ корней;\]
\[2)\ x^{4} + x^{2} - 20 = 0\]
\[\left( x^{2} - 4 \right)\left( x^{2} + 5 \right) = 0\]
\[x_{1,2} = \pm 2;\]
\[3)\ x^{4} + 5x^{2} + 4 = 0\]
\[\left( x^{2} + 1 \right)\left( x^{2} + 4 \right) = 0 \Longrightarrow\]
\[\Longrightarrow корней\ нет;\]
\[4)\ x^{3} - 5x + 4 = 0\]
\[(x - 1)\left( x^{2} + x - 4 \right) = 0\]
\[x^{2} + x - 4 = 0\]
\(D = 1 + 16 = 17\)
\[\Longrightarrow целый\ корень\ только\ \]
\[один:\ \ x = 1.\]
\[Ответ:\ \ 4).\]