Notice that
(1 + x)(1 + y) = 1 + x + y + x y
So we can add 1 to both sides of both equations, and we use the property above to get
a + b + a b = 76 ==> (1 + a)(1 + b) = 77
and
c + d + c d = 54 ==> (1 + c)(1 + d) = 55
Now, 77 = 7*11 and 55 = 5*11, so we get
a + 1 = 7 ==> a = 6
b + 1 = 11 ==> b = 10
(or the other way around, since the given relations are symmetric)
and
c + 1 = 5 ==> c = 4
d + 1 = 11 ==> d = 10
Now substitute these values into the desired quantity:
(a + b + c + d) a b c d = 72,000
