To solve the multiplication of two mixed numbers, \(-1 \frac{5}{9}\) and \(-2 \frac{1}{7}\), we follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
For \(-1 \frac{5}{9}\), convert it to an improper fraction:

\[
-1 \frac{5}{9} = \frac{(9 \times 1) + 5}{9} = \frac{9 + 5}{9} = \frac{14}{9}
\]

So, \(-1 \frac{5}{9} = -\frac{14}{9}\).

For \(-2 \frac{1}{7}\), convert it to an improper fraction:

\[
-2 \frac{1}{7} = \frac{(7 \times 2) + 1}{7} = \frac{14 + 1}{7} = \frac{15}{7}
\]

So, \(-2 \frac{1}{7} = -\frac{15}{7}\).

### Step 2: Multiply the Fractions
Now, multiply the improper fractions:

\[
-\frac{14}{9} \times -\frac{15}{7} = \frac{14 \times 15}{9 \times 7} = \frac{210}{63}
\]

### Step 3: Simplify the Result
The greatest common divisor (GCD) of 210 and 63 is 21. Simplifying the fraction:

\[
\frac{210}{63} = \frac{210 \div 21}{63 \div 21} = \frac{10}{3}
\]

### Step 4: Convert the Result to a Mixed Number
Convert \(\frac{10}{3}\) to a mixed number:

\[
\frac{10}{3} = 3 \frac{1}{3}
\]

So, \(-1 \frac{5}{9} \times -2 \frac{1}{7} = 3 \frac{1}{3}\).

### Final Answer:
\[
3 \frac{1}{3}
\]

Would you like more details on any step or have any questions?

### Related Questions:
1. How do you multiply and divide mixed numbers?
2. What are improper fractions and how are they used in multiplication?
3. Can multiplication of negative mixed numbers be done directly?
4. What is the process for simplifying fractions in more complex cases?
5. How can you find the greatest common divisor (GCD) quickly?

### Tip:
Always convert mixed numbers to improper fractions before performing multiplication or division.

Learn how to multiply the mixed numbers -1 5/9 and -2 1/7 by converting them to improper fractions, multiplying the fractions, simplifying the result, and converting it back to a mixed number. This problem covers key mathematical concepts such as improper fractions, GCD, and fraction simplification. Suitable for students in Grades 6-8.

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