Maximize Area

What's the problem you are trying to solve? The picture doesn't give me any idea of what you're trying to do.

 

Some context would have been nice, such as that the area is the cross-sectional area of what looks to be a trough, formed from sheet metal that is W units wide.

Your picture doesn't convey that information, which would have been helpful.

You have a sign error in this line:

The first '+' sign should be '-'. That makes A_x = WxsinQ - 4xsinQ + 2xsinQcosQ = sinQ[W -4x + 2xcosQ].

The sign error also affects A_Q, which I get as xWcosQ -2x²cosQ + x²cos²Q - x²sin²Q.
About the only thing I can think of to do is to factor x out of each term in this partial.

The critical points are where Q = 0 or when x = 0, either of which gives you an area of 0, and whatever you get from some messy equations that remain. One of these is W - 4x + 2xcosQ = 0, which you can solve for x in terms of Q (and W), or solve for Q in terms of x (and W). Whichever variable you solve for, substitute that in the other equation that comes from A_Q.

 

I didn't work it through that far, so I'll assume that the work you have in post 6 is correct. You can't just "solve the numerators," as you put it, since not all of your expressions have the same denominator; the first has a denominator of 4 - 2cosQ. You'll need to multiply that term by (4 - 2cosQ)/(4 - 2cosQ), and then you can factor out 1/(4 - 2cosQ)² from all four terms. Finally, you'll need to see what values of Q (theta) make A_Q equal to 0.

As I said before, Q = 0 and x = 0 are critical values, but they produce a trough with cross-sectional area 0, so they are not the values you want.

 

Due to the fact that you're working with a physical object, a trough, there is a natural range of values for Q (theta). That should help you eliminate many of the possibilities. If A_xx < 0 and A_QQ < 0, you're at a local maximum.