The domain of the function $f(x) = \sqrt{(25 - x^2)(x + 3)}$ is the set of $x$-values for which the expression inside the square root is non-negative, since the square root of a negative number is not defined in the real number system.

To find the domain, we need to solve the inequality: $(25 - x^2)(x + 3) \geq 0$

Step 1: Find the critical points

We first find where the expression equals zero: $(25 - x^2)(x + 3) = 0$ This occurs when either factor is zero:

• $25 - x^2 = 0$ implies $x = \pm 5$
• $x + 3 = 0$ implies $x = -3$

Thus, the critical points are $x = -5$, $x = -3$, and $x = 5$.

Step 2: Analyze the sign of the product

We will now test the sign of $(25 - x^2)(x + 3)$ in the intervals determined by the critical points: $(-\infty, -5)$, $(-5, -3)$, $(-3, 5)$, and $(5, \infty)$.

1. For $x \in (-\infty, -5)$, both $25 - x^2 < 0$ and $x + 3 < 0$, so the product is positive.
2. For $x \in (-5, -3)$, $25 - x^2 > 0$ and $x + 3 < 0$, so the product is negative.
3. For $x \in (-3, 5)$, $25 - x^2 > 0$ and $x + 3 > 0$, so the product is positive.
4. For $x \in (5, \infty)$, both $25 - x^2 < 0$ and $x + 3 > 0$, so the product is negative.

Step 3: Include the points where the product is zero

At $x = -5$, $25 - x^2 = 0$, so the product is zero. At $x = -3$, $x + 3 = 0$, so the product is zero. At $x = 5$, $25 - x^2 = 0$, so the product is zero.

Step 4: Domain conclusion

The product $(25 - x^2)(x + 3) \geq 0$ in the intervals $(-\infty, -5] \cup [-3, 5]$.

Thus, the domain of $f(x)$ is: $\boxed{(-\infty, -5] \cup [-3, 5]}$

Would you like more details on any of the steps?

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Here are 5 follow-up questions to deepen understanding:

1. How would the domain change if the square root were replaced by a cube root?
2. What is the range of $f(x)$ given the domain you just found?
3. How does the product of two functions determine the sign of their product over an interval?
4. What would the domain be if the function were $f(x) = \sqrt{25 - x^2}$?
5. Can you graph $f(x)$ and mark the domain on the graph?

Tip: Always test the sign of expressions over each interval between critical points to determine the correct domain.