To find the domain of the function $f(x) = \sqrt{-x^3 - 3x^2 + 49x + 147}$, we need to ensure that the expression inside the square root is non-negative, because the square root of a negative number is not defined in the set of real numbers.

Step 1: Set the expression under the square root to be greater than or equal to zero.

$-x^3 - 3x^2 + 49x + 147 \geq 0$ This inequality will define the domain of the function.

Step 2: Factor or analyze the cubic polynomial.

We need to find the roots of the cubic equation: $-x^3 - 3x^2 + 49x + 147 = 0$

Using rational root testing or a graphing method, we can check that one of the real roots is $x = -7$. This allows us to factor the polynomial. We divide $-x^3 - 3x^2 + 49x + 147$ by $(x + 7)$, which gives:

$-x^3 - 3x^2 + 49x + 147 = -(x + 7)(x^2 - 10)$

Thus, we have: $-x^3 - 3x^2 + 49x + 147 = -(x + 7)(x^2 - 10)$

Step 3: Analyze the inequality.

Now, solve: $-(x + 7)(x^2 - 10) \geq 0$

This inequality will hold where the product of these factors is non-negative. We need to find where the sign changes.

• The roots of $x + 7 = 0$ occur at $x = -7$.
• The roots of $x^2 - 10 = 0$ occur at $x = \pm \sqrt{10} \approx \pm 3.162$.

Step 4: Sign analysis.

To determine the intervals where the expression is non-negative, examine the sign of the expression in each interval divided by the roots $x = -7$, $x = -\sqrt{10}$, and $x = \sqrt{10}$.

• For $x < -7$, the expression is negative.
• For $-7 < x < -\sqrt{10}$, the expression is positive.
• For $-\sqrt{10} < x < \sqrt{10}$, the expression is negative.
• For $x > \sqrt{10}$, the expression is positive.

Thus, the function is non-negative in the intervals: $x \in [-7, -\sqrt{10}] \cup [\sqrt{10}, \infty)$

Step 5: Domain.

The domain of the function is: $\boxed{[-7, -\sqrt{10}] \cup [\sqrt{10}, \infty)}$

Do you want further details or have any questions? Here are some related problems to explore:

1. How do you factor cubic polynomials?
2. What are rational root tests and when are they useful?
3. How do you perform sign analysis for polynomial inequalities?
4. How do you solve inequalities involving square roots?
5. How do roots of polynomials relate to the domain of a function?

Tip: When finding the domain of functions involving square roots, always remember to set the expression inside the square root to be greater than or equal to zero, then solve the inequality.