Question

the diagram represents the factorization of ( a^2 + 8a + 12 ). table with columns: empty, ( a ), ?; rows: ( a ) (cell: ( a^2 ), 6a), 2 (cell: 2a, 12) what is the missing number that will complete the factorization? options: 6, 8, 12, 24

Explanation:

Step1: Recall factoring trinomials

To factor \(a^2 + 8a + 12\), we need two numbers that multiply to \(12\) (the constant term) and add up to \(8\) (the coefficient of the middle term). Also, from the table, we see that when we multiply the "?" with \(2\), we get \(12\) (since the last cell is \(12\) and the row has \(2\) and "?"). Let the missing number be \(x\). So \(2\times x = 12\)? Wait, no, looking at the columns: the third column has "?", \(6a\), and \(12\). Also, the first column is \(a\), \(a\), \(2\) (wait, no, the rows: first row is empty, \(a\), "?"; second row is \(a\), \(a^2\), \(6a\); third row is \(2\), \(2a\), \(12\). So in the second row, \(a\times6a\)? No, actually, the table is a rectangle where the product of the first row and first column gives the inner terms. Wait, the trinomial is \(a^2 + 8a + 12\), which factors as \((a + m)(a + n)\) where \(m\times n = 12\) and \(m + n = 8\). From the table, the third column's second row is \(6a\), which is \(a\times6\), and third row is \(12\), which is \(2\times6\). So the missing number (first row, third column) should be \(6\), because \(a\times6 = 6a\) (second row, third column) and \(2\times6 = 12\) (third row, third column). Also, checking the middle term: \(6a + 2a = 8a\), which matches. So the missing number is \(6\).

Step2: Verify with factoring

We factor \(a^2 + 8a + 12\). We need two numbers that multiply to \(12\) and add to \(8\). The numbers are \(6\) and \(2\), since \(6\times2 = 12\) and \(6 + 2 = 8\). So the factorization is \((a + 6)(a + 2)\). Looking at the table, the first row, third column should be \(6\) (since when we multiply \(a\) (first column, second row) by \(6\) (first row, third column), we get \(6a\) (second row, third column), and when we multiply \(2\) (first column, third row) by \(6\) (first row, third column), we get \(12\) (third row, third column)).

Answer:

\(6\) (corresponding to the option "6")