Question

solve.
there is a grassy plain $2x^2 + 20$ft by $3x^2 + 12$ft. there is a walkway with a width of $x$ all the way around the grass.

1. find the area of the grass plus the walkway.
2. find the area of the walkway.

Explanation:

Sub - Question 1:

Step 1: Determine the length and width including the walkway

The grassy plain has dimensions \(2x^{2}+20\) ft (let's say length \(l = 2x^{2}+20\)) and \(3x^{2}+12\) ft (width \(w=3x^{2}+12\)). The walkway has a width of \(x\) all around, so we need to add \(2x\) to both the length and the width ( \(x\) on each side).
New length \(L=(2x^{2}+20)+2x=2x^{2}+2x + 20\)
New width \(W=(3x^{2}+12)+2x=3x^{2}+2x + 12\)

Step 2: Calculate the area of the grass plus walkway

The area \(A\) of a rectangle is \(A = L\times W\). So we multiply the new length and new width:
\[

$$\begin{align*} A&=(2x^{2}+2x + 20)(3x^{2}+2x + 12)\\ &=2x^{2}(3x^{2}+2x + 12)+2x(3x^{2}+2x + 12)+20(3x^{2}+2x + 12)\\ &=6x^{4}+4x^{3}+24x^{2}+6x^{3}+4x^{2}+24x + 60x^{2}+40x + 240\\ &=6x^{4}+(4x^{3}+6x^{3})+(24x^{2}+4x^{2}+60x^{2})+(24x + 40x)+240\\ &=6x^{4}+10x^{3}+88x^{2}+64x + 240 \end{align*}$$

\]

Step 1: Calculate the area of the grassy plain

The area of the grassy plain \(A_{grass}\) is given by the product of its original length and width. So \(A_{grass}=(2x^{2}+20)(3x^{2}+12)\)
\[

$$\begin{align*} A_{grass}&=2x^{2}(3x^{2}+12)+20(3x^{2}+12)\\ &=6x^{4}+24x^{2}+60x^{2}+240\\ &=6x^{4}+84x^{2}+240 \end{align*}$$

\]

Step 2: Calculate the area of the walkway

The area of the walkway \(A_{walkway}\) is the area of the grass plus walkway minus the area of the grass. From part 1, we know the area of grass plus walkway is \(6x^{4}+10x^{3}+88x^{2}+64x + 240\) and the area of grass is \(6x^{4}+84x^{2}+240\)
\[

$$\begin{align*} A_{walkway}&=(6x^{4}+10x^{3}+88x^{2}+64x + 240)-(6x^{4}+84x^{2}+240)\\ &=6x^{4}+10x^{3}+88x^{2}+64x + 240 - 6x^{4}-84x^{2}-240\\ &=10x^{3}+(88x^{2}-84x^{2})+64x+(240 - 240)\\ &=10x^{3}+4x^{2}+64x \end{align*}$$

\]

Answer:

The area of the grass plus the walkway is \(6x^{4}+10x^{3}+88x^{2}+64x + 240\) square feet.

Sub - Question 2: