Question

use the area model to find the product (2a + 5)(2a + 5). first, find the partial products. image of a grid with 2a and 5 as top labels, 2a and 5 as left labels, and four colored rectangles with blanks now, write the product. (2a + 5)(2a + 5) = blank

Explanation:

Step1: Find first partial product (2a×2a)

To find the area of the yellow rectangle, we multiply the length \(2a\) by the width \(2a\). Using the rule of exponents for multiplication (\(x^m\times x^n = x^{m + n}\)) and the coefficient multiplication (\(2\times2 = 4\)), we get:
\(2a\times2a=4a^{2}\)

Step2: Find second partial product (2a×5)

For the teal rectangle, we multiply the length \(2a\) by the width \(5\). The coefficient \(2\) multiplied by \(5\) is \(10\), and we keep the variable \(a\), so:
\(2a\times5 = 10a\)

Step3: Find third partial product (5×2a)

For the orange rectangle, we multiply the length \(5\) by the width \(2a\). Similar to step 2, \(5\times2 = 10\) and we keep the variable \(a\), so:
\(5\times2a=10a\)

Step4: Find fourth partial product (5×5)

For the pink rectangle, we multiply the length \(5\) by the width \(5\). \(5\times5 = 25\), so:
\(5\times5 = 25\)

Step5: Sum the partial products

Now, we add up all the partial products we found: \(4a^{2}+10a + 10a+25\). Combine like terms (\(10a+10a = 20a\)):
\(4a^{2}+20a + 25\)

Answer:

The partial products are \(4a^{2}\), \(10a\), \(10a\), and \(25\). The product \((2a + 5)(2a + 5)=4a^{2}+20a + 25\)