Assume some symmetry.  Place A at (-a,0), E at (a,0), and C at (0,c) and let there be mirror image symmetry about the y-axis.

Assume for the moment that angles at B and D are 180 degrees.

Now we have an isosceles triangle with sides 4, 4, and 2a.

Maximizing this area occurs when a=c= 2√2, the angles at A and E are 45 degrees.  Area = 8.

Second assumption regarding symmetry:

With my hands waving a bit, I'll assume that the pentagon with larger area continues to have the relationship of a=c=x.  Then the pentagon's area will be the area of the same triangle as before, only reduced a little, plus 2 new isosceles triangles ABC and CDE which have sides 2,2, (√2)*x

Area = x^2 + x√(8 - x^2)

Setting the derivative to zero yields:

x = √(2 (2 + √(2))) = approx 2.6131259 which is slightly less than 2√2

Area:  4 + 4√2 = approx 9.656854249 which is a bit more than 8

I can't be 100% sure this is exact since my second assumption about symmetry which may not be valid.