Question

a rectangle has a height of $w^{2}+3w + 9$ and a width of $w^{2}+2$. express the area of the entire rectangle. your answer should be a polynomial in standard form. area =

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is given by $A = \text{height}\times\text{width}$. Here, height $h=w^{2}+3w + 9$ and width $w = w^{2}+2$. So $A=(w^{2}+3w + 9)(w^{2}+2)$.

Step2: Use distributive property

\[

$$\begin{align*} A&=w^{2}(w^{2}+2)+3w(w^{2}+2)+9(w^{2}+2)\\ &=(w^{2}\times w^{2}+w^{2}\times2)+(3w\times w^{2}+3w\times2)+(9\times w^{2}+9\times2) \end{align*}$$

\]

Step3: Simplify each term

\[

$$\begin{align*} w^{2}\times w^{2}&=w^{2 + 2}=w^{4}\\ w^{2}\times2&=2w^{2}\\ 3w\times w^{2}&=3w^{1+2}=3w^{3}\\ 3w\times2&=6w\\ 9\times w^{2}&=9w^{2}\\ 9\times2&=18 \end{align*}$$

\]

Step4: Combine like - terms

\[

$$\begin{align*} A&=w^{4}+2w^{2}+3w^{3}+6w + 9w^{2}+18\\ &=w^{4}+3w^{3}+(2w^{2}+9w^{2})+6w + 18\\ &=w^{4}+3w^{3}+11w^{2}+6w + 18 \end{align*}$$

\]

Answer:

$w^{4}+3w^{3}+11w^{2}+6w + 18$