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The given inequality is:

\[
x^3 + 4x^2 - 11x - 30 < 0
\]

Let's start by factoring the cubic expression.

### Step 1: Try finding rational roots using the Rational Root Theorem.

According to the Rational Root Theorem, possible rational roots are factors of the constant term (-30) divided by factors of the leading coefficient (1), which gives possible roots of:

\[
\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30
\]

Let's test some values of \(x\).

- Test \(x = 2\):

\[
(2)^3 + 4(2)^2 - 11(2) - 30 = 8 + 16 - 22 - 30 = -28 \quad (\text{not a root})
\]

- Test \(x = -2\):

\[
(-2)^3 + 4(-2)^2 - 11(-2) - 30 = -8 + 16 + 22 - 30 = 0 \quad (\text{root found})
\]

Since \(x = -2\) is a root, \(x + 2\) is a factor.

### Step 2: Perform synthetic division to factor out \(x + 2\) from \(x^3 + 4x^2 - 11x - 30\).

Performing synthetic division:

\[
\begin{array}{r|rrrr}
-2 & 1 & 4 & -11 & -30 \\
   &   & -2 & -4  & 30  \\
\hline
   & 1 & 2 & -15  & 0 \\
\end{array}
\]

The quotient is \(x^2 + 2x - 15\).

### Step 3: Factor the quadratic expression \(x^2 + 2x - 15\).

To factor \(x^2 + 2x - 15\), we look for two numbers that multiply to \(-15\) and add up to \(2\). These numbers are \(5\) and \(-3\). So, we factor as:

\[
x^2 + 2x - 15 = (x - 3)(x + 5)
\]

### Step 4: Write the complete factorization.

Thus, the factorization of \(x^3 + 4x^2 - 11x - 30\) is:

\[
x^3 + 4x^2 - 11x - 30 = (x + 2)(x - 3)(x + 5)
\]

### Step 5: Solve the inequality.

We now solve the inequality:

\[
(x + 2)(x - 3)(x + 5) < 0
\]

To solve this, find the critical points where each factor equals zero:

- \(x + 2 = 0 \Rightarrow x = -2\)
- \(x - 3 = 0 \Rightarrow x = 3\)
- \(x + 5 = 0 \Rightarrow x = -5\)

Now, divide the real number line into intervals based on these points: \((- \infty, -5)\), \((-5, -2)\), \((-2, 3)\), and \((3, \infty)\).

Test a point in each interval:

- For \(x = -6\), \((x + 2)(x - 3)(x + 5) = (-)(-)(-) = -\), so \(< 0\).
- For \(x = -3\), \((x + 2)(x - 3)(x + 5) = (-)(-)(+) = +\), so \(> 0\).
- For \(x = 0\), \((x + 2)(x - 3)(x + 5) = (+)(-)(+) = -\), so \(< 0\).
- For \(x = 4\), \((x + 2)(x - 3)(x + 5) = (+)(+)(+) = +\), so \(> 0\).

Thus, the solution to the inequality is:

\[
(-\infty, -5) \cup (-2, 3)
\]

### Final Answer:
The solution to \(x^3 + 4x^2 - 11x - 30 < 0\) is:

\[
x \in (-\infty, -5) \cup (-2, 3)
\]

Would you like further clarification on any part of this? Here are 5 follow-up questions:

1. How do you perform synthetic division step-by-step?
2. What is the Rational Root Theorem and how is it applied?
3. Can all cubic equations be factored this way?
4. How do you test intervals for sign analysis?
5. Why are the critical points important in solving inequalities?

**Tip**: When solving polynomial inequalities, always check the sign of each factor in different intervals determined by the roots.

This tutorial provides a step-by-step solution to the inequality x^3 + 4x^2 - 11x - 30 < 0. Using key algebraic methods like the Rational Root Theorem, synthetic division, and interval testing, we factorize the cubic expression and find the solution set for the inequality. Ideal for students in Grades 10-12.

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