Question

use the area model to find the product (2x + 4)(2x + 7). first, find the partial products. 2x 7 2x 4 now, write the product. (2x + 4)(2x + 7) =

Explanation:

Step1: Find first partial product (2x*2x)

To find the area of the yellow rectangle, we multiply the length \(2x\) by the width \(2x\). Using the rule of exponents for multiplication (\(a^m \cdot a^n=a^{m + n}\)), we have \(2x\times2x=(2\times2)x^{1 + 1}=4x^{2}\).

Step2: Find second partial product (2x*7)

For the teal rectangle, multiply the length \(2x\) by the width \(7\). So, \(2x\times7 = 14x\).

Step3: Find third partial product (4*2x)

For the orange rectangle, multiply the length \(4\) by the width \(2x\). We get \(4\times2x=(4\times2)x = 8x\).

Step4: Find fourth partial product (4*7)

For the pink rectangle, multiply the length \(4\) by the width \(7\). So, \(4\times7=28\).

Step5: Sum the partial products

Now, to find the product of \((2x + 4)(2x+7)\), we sum up all the partial products: \(4x^{2}+14x + 8x+28\). Combine like terms (\(14x+8x = 22x\)), so we have \(4x^{2}+22x + 28\).

Answer:

The partial products are \(4x^{2}\), \(14x\), \(8x\), \(28\) and the product \((2x + 4)(2x + 7)=4x^{2}+22x + 28\)