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 coefficient multiplication. So, \(2x \cdot x^2 = 2 \cdot x^{1+2}=2x^3\).

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

For the 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 \(x\times x=x^2\). So, \(2x\cdot3x = 6x^2\).

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

Multiplying \(2x\) and \(5\), we get \(2x\times5 = 10x\) (we multiply the coefficient \(2\) with \(5\) and keep the variable \(x\) as it is).

Step4: Add the partial products

Now, we add the three partial products together: \(2x^3+6x^2 + 10x\).

Answer:

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