Find the greatest integer value of b for which the expression
has a domain of all real numbers.
If the function has a domain of all real numbers, then dividing by x^2 + bx + 18 always "makes sense".
When does that "make sense"? It is when x^2 + bx + 18 ≠ 0 for any real number x.
Then either x^2 + bx + 18 > 0 for any real number x or x^2 + bx + 18 < 0 for any real number x.
But the latter is not possible, because the graph of y = x^2 + bx + 18 opens upwards.
Then x^2 + bx + 18 > 0 for any real number x.
(i.e., there are no x-intercepts)
Since there are no x-intercepts, \(\Delta = b^2 - 4(1)(18) < 0\).
You can solve the inequality and find the greatest integer b satisfying this inequality.