### Instruction: Find all values of $k$ so that the domain of
\[b(x) = \frac{kx^2 + 2x - 5}{-5x^2 + 2x + k}\]is the set of all real numbers.### Response: I want the domain of $b(x)$ to be all real numbers, which means I don't want any values of $x$ that would make the denominator zero, since that would be undefined.
So I need to find the values of $k$ that would prevent the denominator from being zero, or equivalently, the values of $k$ that would make the equation $-5x^2 + 2x + k = 0$ have no real solutions.
One way to do that is to use the discriminant of a quadratic equation, which is the expression under the square root in the quadratic formula.
If the discriminant is negative, then the equation has no real solutions.
The discriminant of $-5x^2 + 2x + k = 0$ is $2^2 - 4(-5)(k) = 4 + 20k$.
I want this to be negative, so I need $4 + 20k < 0$, or equivalently, $k < -\frac{1}{5}$.