Solution to the Problem:

This can be done by looking for two numbers m and n ,
whose PRODUCT (as Pat says) is 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362880 ,
and whose SUM (Chris adds) is 38² = 1444.
So: m * n = 362880 and m + n = 1444, or n = 1444 - m.

Substituting, we get m(1444 - m) = 362880 ; m² - 1444 m + 362880 = 0 .
Factoring this is the same as looking for two numbers
whose sum is 1444 and product is 362880.

Factor m² - 1444 m + 362880 = 0
(m-1120)(m-324) = 0
m = 1120 and m = 324
Then n = 324 and 1120
Pat and Chris had winnings of $1120 and $324.

And if you want to do it without algebra, make a table of pairs of numbers
that add up to 1444, with their products:
Num 1 Num 2 Product
-------- -------- -------
1 . . . . 1443 . . . 1443
2 . . . . 1442 . . . 2884
3 . . . . 1441 . . . 4323
. . . . . . . . . . . . . . . . . . . . . . . . . .
323 . . . 1121 . . 362083
324 . . . 1120 . . 362880 . . YES!