Find the greatest integer value of $b$ for which the expression $\frac{9x^3+4x^2+11x+7}{x^2+bx+8}$ has a domain of all real numbers.

Consider fraction . The domain of this fraction is .

Find the discriminant of the the equation :

.

There are choices:

1. If D>0, there are two solutions of the equation;

2. D=0, there is only unique solution of the equation;

3. D<0, there are no solutions.

If you should consider case 3, then
.

Therefore, the greatest integer is 5.

step-by-step explanation: