Question

use the box method to distribute and simplify (2x + 4)(-4x - 1). drag and drop the terms to the correct locations of the table. (2x + 4)(-4x - 1)

Explanation:

Step1: Create the box (table)

We have two binomials: \((2x + 4)\) and \((-4x - 1)\). The box method (also known as the area model) for multiplying two binomials \((a + b)(c + d)\) involves creating a 2x2 table. The terms of the first binomial (\(2x\) and \(4\)) are placed along the left - hand side of the table, and the terms of the second binomial (\(-4x\) and \(-1\)) are placed along the top of the table.

	\(-4x\)	\(-1\)
\(4\)	\(4\times(-4x)\)	\(4\times(-1)\)

Step2: Calculate each cell

• For the top - left cell (intersection of \(2x\) and \(-4x\)):

Using the rule of exponents \(a^m\times a^n=a^{m + n}\) and the multiplication of coefficients, we have \(2x\times(-4x)=(2\times(-4))\times(x\times x)=-8x^{2}\)

• For the top - right cell (intersection of \(2x\) and \(-1\)):

\(2x\times(-1)=-2x\)

• For the bottom - left cell (intersection of \(4\) and \(-4x\)):

\(4\times(-4x)=(4\times(-4))x=-16x\)

• For the bottom - right cell (intersection of \(4\) and \(-1\)):

\(4\times(-1)=-4\)

Step3: Combine like terms

Now, we sum up all the terms in the table: \(-8x^{2}-2x - 16x-4\)
Combine the like terms \(-2x\) and \(-16x\): \(-2x-16x=-18x\)

So the simplified form is \(-8x^{2}-18x - 4\)

Answer:

The simplified form of \((2x + 4)(-4x - 1)\) using the box method is \(\boldsymbol{-8x^{2}-18x - 4}\)

And the filled - in table is:

	\(-4x\)	\(-1\)
\(4\)	\(-16x\)	\(-4\)