\[\boxed{\text{676\ (676).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[3x^{2} + bx + 10 = 0,\]
\[\text{\ \ }x_{1} - x_{2} = 4\frac{1}{3} \Longrightarrow x_{1} = \frac{13}{3} + x_{2}\]
\[x^{2} + \frac{b}{3}x + \frac{10}{3} = 0\ \Longrightarrow\]
\[*D = 169 + 120 = 289 = 17^{2}\]
\[x_{1,2} = \frac{- 13 \pm 17}{6} = \frac{2}{3};\ - 5\]
\[Ответ:b = \pm 17.\ \]