Because you seem to have an understanding of the material, I will write my math, but no explanations for my work. Comment on this answer for an explanation, or if I did something wrong.
1. Find the length of the South side of the field.
[tex](5x+2)+(x^2-9)+(x^2-7x+12)\\5x+2+x^2-9+x^2-7x+12\\2x^2+5x+2-9-7x+12\\2x^2-2x+2-9+12\\2x^2-2x+5[/tex]
2.Find the perimeter of the pumpkin field.
[tex]2(5x+2)+2(5x-2)\\(5x+2)+(5x+2)+(5x-2)+(5x-2)\\20x+4-4\\20x[/tex]
3. Find the perimeter of the whole field.
[tex]2(4x+5x-2)+2(5x+2+x^2-9+x^2-7x+12)\\2(9x-2)+2(2x^2-2x-5)\\18x-4+4x^2-4x+10\\4x^2+14x+6[/tex]
4. If [tex]x=30ft[/tex], what is the total perimeter?
[tex]P=4x^2+14x+6\\P=4(30)^2+14(30)+6\\P=4(900)+420+6\\P=1026ft[/tex]
Random Tangent:
If you want to involve [tex]y[/tex] in finding these answers for whatever reason, your math is going to get messy. Here's what I tried before I realized you could multiply each side by [tex]2[/tex] to get the total perimeter:
[tex]xy+1+x+6=4x+5x-2\\xy-8x=-9\\x(y-8)=-9\\y-8=-\frac{9}{x}\\y=-\frac{9}{x}+8[/tex]
Now the other side. This one's even worse.
[tex]xy-1+6x+x=5x+2+x^2-9+x^2-7x+12\\xy+9x-2x^2=5\\x(y+9+2x)=5\\2x+9+y=\frac{5}{x}\\y=\frac{5}{x}-2x-9[/tex]
At this point, was realizing I wasn't going about the problem correctly, but did one last try by setting [tex]y=y[/tex]. At this point in the equation, I simply had to stop, both due to complexity and technological problems.
[tex]\frac{5}{x}-2x-9=-\frac{9}{x}+8\\\frac{14}{x}=2x+17\\ 14=2x^2+17x\\0=2x^2+17x-13\\[/tex]
I could possibly continue to solve, but I don't see a point. It was clear that this was NOT the right path to take finding the perimeter. It was here I realized that I could just multiply the two sides without [tex]y[/tex] in them by [tex]2[/tex] and--without as much difficulty--reach the correct answer.
