Notice that

uvx + uvy + uwx + uwy = u(x+y)(v+w).

If  u, v, w, x, y are allowed to be negative, there is no largest value, since, for example, 
one could let  u, x, y be arbitrarily “large” negative numbers (so  u(x+y) is positive), 
then  v+w is a large positive number and the whole product can be as large as you like.

If they are restricted to positive real numbers, then from AM-GM, we have

     >= 


Using the imposed condition to the sum, the left side is just  5, so we have found

    } <=  = 125.


The maximum value is  125, and occurs if, and only if,  u = x+y = v+w = 5.