Since $x$ is the length (in feet) of the short side of the rectangular field, then the area of the rectangular field is $2x^2$, and its perimeter is $6x$. Thus, the side length of the square field is $(600-6x)/4=150-\frac32x$, so its area is $22500-450x+\frac94x^2$, and the total area is $$A(x)=22500-450x+\frac{17}4x^2,$$ as desired.

We know in particular that we need $2x^2\geq 800$, from which $x\geq 20$ (since $x$ is positive). Also, since the square field needs area at least $100$ square feet, then its side length is at least $10$ feet, meaning $150-\frac32x\geq 10,$ from which $x\leq\frac{280}3$, as desired.

As for the last part, observe that the graph of $A(x)$ is a portion of a parabola that opens up. This will necessarily be maximized at one of its endpoints (if it opened down, it might maximize at its vertex, if in the domain). It is easy to check which of $x=20$ or $x=\frac{280}3$ maximizes $A(x)$, and what that maximum area is.

Two Field Problem
I'm using the latest Firefox without Adobe Flash (so HTML5 videos) on Youtube

Youtube videos almost always start kind of black, parts of the image are barely visible.

Why do I have to wait a few seconds for the image to "stabilize" and be able to watch the video ?
Is there something I can do to have the picture/video load instantly ?

Here is what is happening in two pictures (timeline is wrong, I just clicked back to refesh the picture) :




