Questions: Multiply. 2 x^2(x+3)^2 2 x^2(x+3)^2= (Simplify your answer.)

Transcript text: Multiply. \[ \begin{array}{c} 2 x^{2}(x+3)^{2} \\ 2 x^{2}(x+3)^{2}= \end{array} \] (Simplify your answer.)

Solution

Solution Steps

To simplify the expression \(2x^2(x+3)^2\), we need to expand the binomial \((x+3)^2\) first, and then multiply the result by \(2x^2\). The binomial expansion \((x+3)^2\) can be calculated using the formula \((a+b)^2 = a^2 + 2ab + b^2\). After expanding, distribute \(2x^2\) across the resulting polynomial.

Step 1: Expand the Binomial

We start with the expression \(2x^2(x+3)^2\). First, we expand the binomial \((x+3)^2\): \[ (x+3)^2 = x^2 + 2 \cdot 3 \cdot x + 3^2 = x^2 + 6x + 9 \]

Step 2: Distribute \(2x^2\)

Next, we distribute \(2x^2\) across the expanded binomial: \[ 2x^2(x^2 + 6x + 9) = 2x^2 \cdot x^2 + 2x^2 \cdot 6x + 2x^2 \cdot 9 \] This results in: \[ 2x^4 + 12x^3 + 18x^2 \]