\(x + y+ xy = 39\)
\(x( 1 + y) + y = 39\)
\(x(1+y) + 1 + y = 39 + 1\)
\(x(1+y) + 1(1+y) = 40\)
\((x+1)(y+1)=40\)
The only pairs that work are \((0,39)\), \((1, 19)\), \((3, 9)\), and \((4, 7)\)
The product of sums and products are 0, 380, 324, and 308.
So, the largest possible value is \(\color{brown}\boxed{380}\)
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