To find the domain of the function $f(x) = \frac{2x + 5}{\sqrt{-x^2 - 4x + 21}}$, we need to determine the values of $x$ for which the function is defined. Specifically, the denominator involves a square root, and the expression inside the square root must be non-negative because the square root of a negative number is not real.

Step-by-step analysis:

1. The function is undefined if the denominator is zero or negative. Therefore, we must have: $-x^2 - 4x + 21 \geq 0$ This is a quadratic inequality. Let’s solve this inequality to find the domain.

2. First, solve the corresponding quadratic equation: $-x^2 - 4x + 21 = 0$ We can multiply the entire equation by $-1$ to simplify: $x^2 + 4x - 21 = 0$ Now, solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 4$, and $c = -21$. Substituting these values into the formula gives: $x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-21)}}{2(1)}$ $x = \frac{-4 \pm \sqrt{16 + 84}}{2}$ $x = \frac{-4 \pm \sqrt{100}}{2}$ $x = \frac{-4 \pm 10}{2}$ So, the two solutions are: $x = \frac{-4 + 10}{2} = 3 \quad \text{and} \quad x = \frac{-4 - 10}{2} = -7$

3. These are the critical points where the quadratic expression equals zero. To determine the intervals where the quadratic expression is non-negative, we can test values in the intervals $(- \infty, -7)$, $(-7, 3)$, and $(3, \infty)$.

   • For $x = -8$ (a point in $(- \infty, -7)$): $-(-8)^2 - 4(-8) + 21 = -64 + 32 + 21 = -11 \quad (\text{negative})$
   • For $x = 0$ (a point in $(-7, 3)$): $-(0)^2 - 4(0) + 21 = 21 \quad (\text{positive})$
   • For $x = 4$ (a point in $(3, \infty)$): $-(4)^2 - 4(4) + 21 = -16 - 16 + 21 = -11 \quad (\text{negative})$

4. Therefore, the quadratic expression is non-negative on the interval $[-7, 3]$.

Conclusion:

The function $f(x)$ is defined when the expression under the square root is non-negative, which happens for $x \in [-7, 3]$. Therefore, the domain of $f(x)$ is: $\boxed{[-7, 3]}$

Would you like more details, or do you have any questions?

Here are five related questions:

1. How do you solve inequalities involving square roots?
2. What is the significance of the discriminant in the quadratic formula?
3. How would the domain change if the denominator had no square root?
4. Can you explain how to test intervals in more detail?
5. What is the domain of a function without any denominators or square roots?

Tip: When solving inequalities involving quadratics, remember to check the intervals between the roots to determine where the expression is positive or negative.