What is $\frac{-5}{9}\cdot \frac{8}{17}$?
Recall that if $b$ and $d$ are nonzero, then  $$\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}.$$That is, to multiply fractions, we multiply the numerators and multiply the denominators.  The product of the given numerators is $-5\cdot 8=-40$.  The product of the denominators is $9\cdot 17=153$.  So,  $$\frac{-5}{9}\cdot \frac{8}{17}=\frac{-5\cdot 8}{9\cdot 17}=\frac{-40}{153}=\boxed{-\frac{40}{153}}.$$