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.