Question

use the area model to find the product $4p(p^{2} + 5p + 2)$. first, find the partial products. now, write the product. $4p(p^{2} + 5p + 2) = \boxed{\quad}$

Explanation:

Step1: Recall Distributive Property

The distributive property states that \(a(b + c + d)=ab + ac + ad\). Here, \(a = 4p\), \(b=p^{2}\), \(c = 5p\), \(d = 2\).

Step2: Sum the Partial Products

From the area model, the partial products are \(4p\times p^{2}=4p^{3}\), \(4p\times5p = 20p^{2}\), and \(4p\times2=8p\). Now, sum these partial products: \(4p^{3}+20p^{2}+8p\).

Answer:

\(4p^{3}+20p^{2}+8p\)