Question 137160
You can set up a table of two columns, first column is x values and second column is the function P = -x^2 + 12x + 44.
Plug in x values such that x = 0, 1, 2, ... (the least number of hotdog John can sell is none, so start at 0) and find P. 
For example, for x = 0, P = 44; x = 1, P = 55; and so on.
You will see that when you plug in x = 6, P = 80; but when x = 7, P = 79. So the profit decreases when x >= 7. Therefore the most profit John can earn is $80, when 6 hotdogs are sold.