8-53 Fill in the product and sum based on the table provided.

Steps to Solve

1. Identify the Algebraic Terms

The table presents specific algebraic expressions. We have terms:

• In the first row:

  • The first term: $x$
  • The second term: $-3x$
  • The third term: $-6y$
  • The fourth term: $12$

• In the second row:

  • The first term: $2x^2$
  • The second term: $4xy$
  • The third term: $-8x$

2. Calculate the Product

The product can be calculated by multiplying the terms in the columns.

• For the first column: $$ x \cdot (-3x) = -3x^2 $$

• For the second column: $$ -6y \cdot 4xy = -24xy $$

• For the third column: $$ 12 \cdot (-8x) = -96x $$

Now, combine these results for the final product:

$$ \text{Product} = -3x^2 - 24xy - 96x $$

3. Calculate the Sum

The sum can be calculated by adding the terms in the columns.

• For the first column: $$ x + (-3x) = -2x $$

• For the second column: $$ -6y + 4xy = 4xy - 6y $$

• For the third column: $$ 12 + (-8x) = 12 - 8x $$

Now, combine these results for the final sum:

$$ \text{Sum} = -2x + (4xy - 6y) + (12 - 8x) $$

4. Final Simplification

Combine like terms for the sum obtained:

• The coefficient of $x$: $$ -2x - 8x = -10x $$

Combine it as:

$$ \text{Final Sum} = -10x + 4xy - 6y + 12 $$