The quadratic x^2+440x+440^2 can be written in the form (x+b)^2+c, where b and c are constants. what is c/b?

This form (x+b)²+c is a vertex form that can be found by completing square for the equation x²+440x+440².

To complete square we will need to use formula  a² +2ab + b² = (a+b)²

 x²+440x+440²,
x² +2*220²x +220² -220² +440²  (we added 220² -220², so nothing really changed here).

x² +2*220²x +220² -220² +440² ( here , bold part can be rewritten as (x+220)²)

(x+220)² -220² +440² (calculating bold part)
 
(x+220)² -220² +440² = (x+220)²  + 145200

Now we have  x²+440x+440², in the form that we need 
 x²+440x+440² = (x+220)²  + 145200

(x+b)²+c = (x+220)²  + 145200,
b=220,
c= 145200

So, 
c/b = 145200/220 = 660
c/b = 660/1 (if you need it in a fraction form)  or 660.