There you go man, look at your original post you have said it is (f(x)+f(-x)) (g(x)+g(-x))

 

Well if the integrand is h(x) = (f(x)+f(-x)) (g(x)-g(-x))

 

Then h(-x) = (f(-x)+f(x)) (g(-x)-g(x))

               = -(g(x)-g(-x)) (f(x)+f(-x))

              = -h(x)

Thus it is established that h(x) is an odd function.