Question

which terms complete the factorization of ( x^2 + 27x + 162 ) represented by the model?
grid with cells: top row has x in first cell, middle row has x in first cell, x² in second; bottom row has 18 in first cell, 162 in third
options:

• 27, 9x, 18x
• 9, 9x, 18x
• 27, 9x, 27x
• 9, 9x, 27x

Explanation:

Step1: Recall factoring trinomials

To factor \(x^{2}+27x + 162\), we need two numbers that multiply to \(162\) and add up to \(27\). Let's find the factors of \(162\): \(1\times162 = 162\), \(2\times81=162\), \(3\times54 = 162\), \(6\times27=162\), \(9\times18 = 162\). Among these, \(9\) and \(18\) add up to \(27\) (\(9 + 18=27\)). So, \(x^{2}+27x + 162=(x + 9)(x + 18)\).

Step2: Analyze the area model

In the area model (rectangle with sub - rectangles), the area of the big rectangle is \(x^{2}+27x + 162\). The top - left cell is \(x\) (from \((x + 9)\) or \((x + 18)\)), the middle - left cell is \(x\) and the middle - top cell is \(x^{2}\) (which is \(x\times x\)). The bottom - left cell is \(18\) and the bottom - right cell is \(162\) (which is \(18\times9\) or \(18\times18\)? Wait, no, from the factoring \((x + 9)(x + 18)=x^{2}+18x+9x + 162=x^{2}+27x + 162\). So the missing cells: the top - right cell (next to \(x\)) should be \(9\) (since one factor is \(x + 9\)), the middle - right cell (next to \(x^{2}\)) should be \(9x\) (since \(x\times9 = 9x\)), and the bottom - middle cell (next to \(18\)) should be \(18x\) (since \(18\times x=18x\))? Wait, no, let's re - express the area model. The length and width of the big rectangle are \((x + a)\) and \((x + b)\) where \(a\) and \(b\) are the constants. The area is \(x^{2}+(a + b)x+ab\). Here, \(ab = 162\) and \(a + b=27\), so \(a = 9\), \(b = 18\) (or vice - versa). So the top row: first cell \(x\), second cell \(9\) (since width is \(x + 9\)). The middle row: first cell \(x\), second cell \(x\times9=9x\). The bottom row: first cell \(18\), second cell \(18\times x = 18x\), third cell \(18\times9 = 162\). So the missing terms are \(9\) (top - right), \(9x\) (middle - right), \(18x\) (bottom - middle). Wait, but looking at the options, the second option is \(9,9x,18x\). Let's check the options again. The options are:

• Option 1: \(27,9x,18x\)
• Option 2: \(9,9x,18x\)
• Option 3: \(27,9x,27x\)
• Option 4: \(9,9x,27x\)

From the factoring \((x + 9)(x + 18)=x^{2}+18x + 9x+162=x^{2}+27x + 162\). In the area model, the top - right cell (the one next to \(x\) in the top row) should be \(9\) (because the factor is \(x + 9\)), the middle - right cell (next to \(x^{2}\)) is \(9x\) (since \(x\times9 = 9x\)), and the bottom - middle cell (next to \(18\)) is \(18x\) (since \(18\times x=18x\)). So the terms that complete the factorization are \(9\), \(9x\), and \(18x\).

Answer:

B. 9, 9x, 18x