(In reply to
Solution by Brian Smith)
Your solution is remarkable similar to my approach: try linear, then quadratic. But near the end you state:
Therefore for f(x) to have a factorization into two quadratic
expressions there must be a factorization of 63 whose two terms sum to a
square number. There are in fact two ways: 1 and 63 or 7 and 9. In
the first case f(x) has a positive M of 64, and in the second case
positive M is 16.
For your solution the second case corresponds to f+g = 16 = a^2
Then from the linear terms of your factorization m = afag = 8
so m=8
(and in your first case m=496)

Posted by Jer
on 20160403 20:37:25 