Question

intro to box method (generic rectangles)
score: 0/3 penalty: none
question
what is the missing term?
-5 ? 10
x 5x² -2x
5x -2

Explanation:

Step1: Recall box method multiplication

In the box method, the product of the terms on the side (row and column) gives the term in the box. So, for the first row (with \(-5\)) and first column (with \(5x\)), the missing term is \(-5\times5x\).

Step2: Calculate the product

Multiply \(-5\) and \(5x\): \(-5\times5x = -25x\). Wait, no, wait. Wait, the first column is \(5x\) (bottom left is \(5x^2\) from \(x\times5x\), so first column is \(5x\), first row is \(-5\). Wait, bottom left is \(x\times5x = 5x^2\), so first column is \(5x\), second column is \(-2\) (since \(x\times(-2)= -2x\)). Then first row, first column: \(-5\times5x\)? Wait no, wait the first row is \(-5\), first column is \(5x\), so the box should be \(-5\times5x\)? Wait no, wait the bottom left is \(x\times5x = 5x^2\), so the first column is \(5x\), second column is \(-2\) (because \(x\times(-2)= -2x\)). Then the top left box: row is \(-5\), column is \(5x\), so \(-5\times5x = -25x\)? Wait no, that can't be. Wait, maybe I mixed up. Wait, the bottom row is \(x\), so bottom left is \(x\times5x = 5x^2\), bottom right is \(x\times(-2)= -2x\). Top row is \(-5\), so top right is \(-5\times(-2)=10\), which matches. So top left should be \(-5\times5x\)? Wait, no, \(5x\) is the first column (since bottom left is \(x\times5x\), so column 1 is \(5x\), row 1 is \(-5\), so top left is \(-5\times5x = -25x\)? Wait, but let's check again. Wait, the columns are \(5x\) and \(-2\) (because \(5x\times x = 5x^2\), \(-2\times x = -2x\)). Rows are \(-5\) and \(x\). So top left: \(-5\times5x = -25x\)? Wait, but maybe I made a mistake. Wait, no, the top right is \(-5\times(-2)=10\), which is correct. Bottom left is \(x\times5x = 5x^2\), bottom right is \(x\times(-2)= -2x\), correct. So top left is \(-5\times5x = -25x\)? Wait, but let's see: the box method is (row1 + row2)(col1 + col2) = row1col1 + row1col2 + row2col1 + row2col2. So ( -5 + x)(5x - 2) = (-5)(5x) + (-5)(-2) + x(5x) + x(-2) = -25x + 10 + 5x^2 - 2x. So the top left is -25x. Wait, but the question's top left is "?", so that should be -25x? Wait, no, wait the bottom left is 5x², which is x5x, so col1 is 5x, row1 is -5, so col1row1 = -55x = -25x. Yes, that makes sense. So the missing term is -25x? Wait, but let's check again. Wait, the bottom row is x, so x times col1 is 5x², so col1 is 5x² / x = 5x. Col2: x times col2 is -2x, so col2 is -2x / x = -2. Then row1 is -5, so row1 times col1 is -55x = -25x, row1 times col2 is -5(-2)=10, which matches the top right. So the missing term is -25x. Wait, but maybe I messed up. Wait, no, that's the box method. So the missing term is -25x? Wait, but let's check the numbers. Wait, 5x² is x5x, so col1 is 5x, row2 is x. Then row1 is -5, so row1col1 is -55x = -25x. Yes. So the missing term is -25x.

Answer:

\(-25x\)