The sum of the product and the sum of two positive integers is 39. Find the largest possible value of the product of their sum and their product.
+2668 \(x + y+ xy = 39\)
\(x( 1 + y) + y = 39\)
\(x(1+y) + 1 + y = 39 + 1\)
\(x(1+y) + 1(1+y) = 40\)
\((x+1)(y+1)=40\)
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 \(\color{brown}\boxed{380}\)
