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Negative numbers can be tricky, but understanding them is important. Think of negative numbers as the opposite of positive numbers. When you multiply two negative numbers, the result is always positive. This is because two negatives cancel each other out. For example, \((-7) \times (-8)\,=\,56\). However, when you multiply a positive number by a negative number, the result is negative. This is why \[56 \times (-9)\,=\,-504\]. Negative numbers follow specific rules. Remember these rules, and you will multiply negative numbers correctly.

Understanding the multiplication rules for negative and positive numbers is crucial. Here's a quick summary to help you:

• Negative \( \times \) Negative = Positive
• Positive \( \times \) Negative = Negative
• Positive \( \times \) Positive = Positive

Use these rules to multiply any integers. In our exercise, we see these rules in action:

\[(-7) \times (-8) = 56\]
\[56 \times (-9) = -504\]
\[-504 \times (-10) = 5040\]
Practicing these rules will help you become more comfortable with multiplying integers, especially when negative numbers are involved.

Let's break down the original problem into step-by-step solutions to make it easier to understand:

1. Multiply the first two numbers:
Start with \( -7 \times -8 = 56 \). The product is positive because both numbers are negative.

2. Multiply the result by the third number:
Take the result from Step 1, which is 56, and multiply it by -9:
\[-56 \times (-9) = -504\]. The result is negative because a positive number times a negative number is negative.

3. Multiply by the fourth number:
Finally, multiply -504 by -10: \[-504 \times (-10) = 5040\]. The result is positive as multiplying two negatives gives a positive.

Following these clear steps systematically helps solve complex multiplication problems involving negative numbers. Always be mindful of the multiplication rules throughout the process.