Let f(x) = (2x^2 + x + 5)/(x^2 + 6x + c).
Find the smallest integer value of c so that f(x) has a domain of all real numbers.
The numerator is defined for all real numbers
We need to have complex roots for the polynomial in the denominator
This will happen when the discriminant is < 0 ....so....
6^2 - 4c < 0
36 - 4c < 0
36 < 4c
9 < c means the same thing as c < 9
So....when c = 10, this rational function will have a domain of all reals
See the graph here to confirm this : https://www.desmos.com/calculator/mkpngao0r3