+193 What is the smallest integer value of $c$ such that the function $f(x)=\frac{2x^2+x+5}{x^2+4x+c}$ has a domain of all real numbers?
+128473 To have a domain of all real numbers, the denominator cannot evaluate to 0 for any real x
Therefore.....the discriminant must be < 0 for the polynomial in the denominator
So
4^2 - 4 (1)c < 0
16 - 4c < 0
16 < 4c
16/4 < c
c > 4
Therefore c = 5 is the smallest integer value of c that guarantees a domain of all reals
