To solve the problem multiplying the mixed number [tex]\( 5 \frac{1}{6} \)[/tex] and the fraction [tex]\( -\frac{2}{5} \)[/tex], we follow these steps:
1. Convert the mixed number to an improper fraction:
[tex]\[
5 \frac{1}{6} = \frac{5 \cdot 6 + 1}{6} = \frac{30 + 1}{6} = \frac{31}{6}
\][/tex]
2. Multiply the two fractions:
[tex]\[
\frac{31}{6} \cdot \left(-\frac{2}{5}\right) = \frac{31 \cdot (-2)}{6 \cdot 5} = \frac{-62}{30}
\][/tex]
3. Simplify the fraction if possible:
To simplify [tex]\(\frac{-62}{30}\)[/tex], we find the greatest common divisor (GCD) of 62 and 30, which is 2:
[tex]\[
\frac{-62 \div 2}{30 \div 2} = \frac{-31}{15}
\][/tex]
So, the final answer is:
[tex]\[
\boxed{-20958489923773395135801964031727/10141204801825835211973625643008}
\][/tex]
