Question

the diagram represents the factorization of $a^2 + 8a + 12$. what is the missing number that will complete the factorization? options: 6, 8, 12, 24 (with a grid diagram showing rows and columns with terms like $a$, $a^2$, $6a$, $2$, $2a$, $12$ and a ?)

Explanation:

Step1: Analyze the table structure

In the factorization table (area model), the product of the terms in the same row or column should match the polynomial. Let the missing number be \( x \). The third column has terms \( 6a \), \( 12 \), and the top term \( x \) (wait, no, the first row third column is \( ? \), second row third is \( 6a \), third row third is \( 12 \). Also, the third row: \( 2 \times a = 2a \), \( 2 \times ? = 12 \)? Wait, no, let's think about the columns. The second column: \( a \times a = a^2 \), \( a \times 2 = 2a \). The third column: \( ? \times a = 6a \), \( ? \times 2 = 12 \). Let's solve \( ? \times 2 = 12 \), so \( ? = \frac{12}{2}=6 \). Let's check with the other product: \( ? \times a = 6a \), if \( ? = 6 \), then \( 6 \times a = 6a \), which matches. Also, the polynomial is \( a^2 + 8a + 12 \), factored as \( (a + 6)(a + 2) \), so the missing term should be 6.

Step2: Verify the factorization

The area model represents \( (a + 2)(a + 6) \), since \( a \times a = a^2 \), \( a \times 6 = 6a \), \( 2 \times a = 2a \), \( 2 \times 6 = 12 \). Adding the middle terms: \( 6a + 2a = 8a \), so the polynomial is \( a^2 + 8a + 12 \), which matches. So the missing number is 6.

Answer:

6 (corresponding to the option "6")