### 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: To find the domain of a rational function, I need to avoid values of $x$ that would make the denominator zero.
So I want to find the values of $k$ that make the denominator never zero, or in other words, have no real roots.
I can use the discriminant of the quadratic equation to determine when a quadratic has no real roots.
The discriminant is the part under the square root in the quadratic formula, $b^2 - 4ac$.
If the discriminant is negative, then the quadratic has no real roots.
So I want to find the values of $k$ that make the discriminant of the denominator negative.
The denominator is $-5x^2 + 2x + k$, so its discriminant is $(2)^2 - 4(-5)(k) = 4 + 20k$.
I want this to be negative, so I need to solve $4 + 20k < 0$.
I can divide both sides by 4 to get $1 + 5k < 0$.
Then I can subtract 1 from both sides to get $5k < -1$.
Finally, I can divide both sides by 5 to get $k < -\frac{1}{5}$.
This means that any value of $k$ that is less than $-\frac{1}{5}$ will make the domain of $b(x)$ the set of all real numbers.