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The table you provided asks for calculations using the values of \( a \) and \( b \) in different expressions. Let's work through the table and fill in the missing values step by step.

The expressions to be calculated are as follows:
1. \( ab \)
2. \( (-a)b \)
3. \( -(ab) \)
4. \( a(-b) \)
5. \( (-a)(-b) \)

### First row: \( a = -2 \) and \( b = 6 \)
1. \( ab = (-2) \times 6 = -12 \)
2. \( (-a)b = (2) \times 6 = 12 \)
3. \( -(ab) = -(-12) = 12 \)
4. \( a(-b) = (-2) \times (-6) = 12 \)
5. \( (-a)(-b) = (2) \times (-6) = -12 \)

### Second row: \( a = 3 \) and \( b = -2.5 \)
1. \( ab = 3 \times (-2.5) = -7.5 \) (already given)
2. \( (-a)b = (-3) \times (-2.5) = 7.5 \)
3. \( -(ab) = -(-7.5) = 7.5 \)
4. \( a(-b) = 3 \times (2.5) = 7.5 \)
5. \( (-a)(-b) = (-3) \times (2.5) = -7.5 \)

### Third row: \( a = 3 \) and \( b = -5 \)
1. \( ab = 3 \times (-5) = -15 \)
2. \( (-a)b = (-3) \times (-5) = 15 \) (already given)
3. \( -(ab) = -(-15) = 15 \)
4. \( a(-b) = 3 \times (5) = 15 \)
5. \( (-a)(-b) = (-3) \times (5) = -15 \)

### Fourth row: \( a = 8 \) and \( b = 5 \)
1. \( ab = 8 \times 5 = 40 \) (already given)
2. \( (-a)b = (-8) \times 5 = -40 \)
3. \( -(ab) = -(40) = -40 \)
4. \( a(-b) = 8 \times (-5) = -40 \)
5. \( (-a)(-b) = (-8) \times (-5) = 40 \)

### Completed Table:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
a & b & ab & (-a)b & -(ab) & a(-b) & (-a)(-b) \\
\hline
-2 & 6 & -12 & 12 & 12 & 12 & -12 \\
3 & -2.5 & -7.5 & 7.5 & 7.5 & 7.5 & -7.5 \\
3 & -5 & -15 & 15 & 15 & 15 & -15 \\
8 & 5 & 40 & -40 & -40 & -40 & 40 \\
\hline
\end{array}
\]

Let me know if you'd like more details or clarifications. Here are some follow-up questions to test your understanding:

1. What happens to the value of \( ab \) when both \( a \) and \( b \) are negative?
2. How does \( (-a)b \) compare to \( ab \)?
3. Why is \( -(ab) \) always the opposite of \( ab \)?
4. How do the expressions \( a(-b) \) and \( (-a)(-b) \) relate to each other?
5. If \( a \) or \( b \) is 0, what would the table look like?

**Tip:** When multiplying two negative numbers, the result is always positive.

The user submitted a table asking to fill in values of various algebraic expressions like ab, (-a)b, -(ab), and a(-b) for different values of a and b.

Complete a table of algebraic expressions involving two variables, a and b, where students learn how to multiply integers and apply the rules of signs. This problem demonstrates how to handle negative values in multiplication and helps reinforce key algebraic concepts.

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