Recall that division of fractions is the same as multiplying the divident by the reciprocal of the divisor which means that
[tex]\frac{5}{6}\div-\frac{9}{8}\Longrightarrow\frac{5}{6}\times-\frac{8}{9}[/tex]Proceed with the multiplication and we get
[tex]\begin{gathered} \frac{5}{6}\times-\frac{8}{9} \\ =\frac{5}{6}\times-\frac{8}{9} \\ =\frac{5\times-8}{6\times9} \\ =\frac{-40}{54} \\ \\ \text{reduce the answer to its simplest form} \\ \frac{-40}{54} \\ =-\frac{20}{27} \\ \\ \text{Therefore,} \\ \frac{5}{6}\div-\frac{9}{8}=-\frac{20}{27} \end{gathered}[/tex]