Where's that function notation you studied earlier? You need it now.
Area is a function of x and y
Area(x,y) = x*y
We also know that x and y are related, so y can be expressed in terms of x.
2x + 2y = 600
x + y = 300
y = 300 - x
After substitution, this gives
Area(x) = x*(300-x) = 300x - x^2
This should look like a nice quadratic or parabola. You should have studied how to find the maximum of such things.
Show us what you get.