Question

use the area model to find the product $2x(x^2 + 3x + 5)$.
first, find the partial products.
now, write the product.
$2x(x^2 + 3x + 5) = \square$

Explanation:

Step1: Multiply \(2x\) by \(x^2\)

To find the partial product of \(2x\) and \(x^2\), we use the rule of exponents for multiplication (\(a^m \cdot a^n = a^{m + n}\)) and the commutative property of multiplication. So, \(2x \cdot x^2 = 2 \cdot x^{1+2}=2x^3\).

Step2: Multiply \(2x\) by \(3x\)

For the partial product of \(2x\) and \(3x\), we multiply the coefficients and the variables separately. The coefficient part is \(2\times3 = 6\), and the variable part is \(x\times x=x^{1 + 1}=x^2\). So, \(2x\cdot3x = 6x^2\).

Step3: Multiply \(2x\) by \(5\)

To find the partial product of \(2x\) and \(5\), we multiply the coefficient \(2\) with \(5\) and keep the variable \(x\) as it is. So, \(2x\cdot5=10x\).

Step4: Combine the partial products

Now, we add up all the partial products we found in the previous steps. So, \(2x(x^2 + 3x + 5)=2x^3+6x^2 + 10x\).

Answer:

\(2x^3 + 6x^2 + 10x\)