To solve the inequality \( x^3 + x^2 > 9x + 9 \), follow these steps:

### Step 1: Rewrite the inequality
First, bring all terms to one side of the inequality:

\[
x^3 + x^2 - 9x - 9 > 0
\]

Now, we need to solve the inequality \( x^3 + x^2 - 9x - 9 > 0 \).

### Step 2: Factor the cubic expression
Let's attempt factoring by grouping or other methods. We try factoring this expression:

\[
x^3 + x^2 - 9x - 9 = (x^2 - 9)(x + 1)
\]

### Step 3: Factor further
Notice that \( x^2 - 9 \) is a difference of squares, so it can be factored further:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

Thus, the factored form of the cubic polynomial is:

\[
(x - 3)(x + 3)(x + 1) > 0
\]

### Step 4: Solve the inequality
Now, we need to solve \( (x - 3)(x + 3)(x + 1) > 0 \). This is a product of three factors, so we analyze where the expression changes signs. The critical points are \( x = 3 \), \( x = -3 \), and \( x = -1 \).

We can determine the sign of the expression in the intervals divided by these critical points: \( (-\infty, -3) \), \( (-3, -1) \), \( (-1, 3) \), and \( (3, \infty) \).

### Step 5: Test intervals
- For \( x \in (-\infty, -3) \), choose \( x = -4 \):
  \[
  (-4 - 3)(-4 + 3)(-4 + 1) = (-7)(-1)(-3) = -21 \quad (\text{negative})
  \]
- For \( x \in (-3, -1) \), choose \( x = -2 \):
  \[
  (-2 - 3)(-2 + 3)(-2 + 1) = (-5)(1)(-1) = 5 \quad (\text{positive})
  \]
- For \( x \in (-1, 3) \), choose \( x = 0 \):
  \[
  (0 - 3)(0 + 3)(0 + 1) = (-3)(3)(1) = -9 \quad (\text{negative})
  \]
- For \( x \in (3, \infty) \), choose \( x = 4 \):
  \[
  (4 - 3)(4 + 3)(4 + 1) = (1)(7)(5) = 35 \quad (\text{positive})
  \]

### Step 6: Write the solution
The inequality \( (x - 3)(x + 3)(x + 1) > 0 \) is satisfied where the product is positive, which occurs in the intervals \( (-3, -1) \) and \( (3, \infty) \).

Thus, the solution to the inequality is:

\[
x \in (-3, -1) \cup (3, \infty)
\]

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

### Related Questions:
1. How do you solve a cubic inequality graphically?
2. What are the methods to factor cubic polynomials?
3. How do you determine the sign of polynomial expressions across intervals?
4. Can cubic inequalities have multiple solutions?
5. How does the Intermediate Value Theorem apply to polynomial inequalities?

### Tip:
Always check the sign of each interval carefully when solving polynomial inequalities.

Learn how to solve the cubic inequality x^3 + x^2 > 9x + 9 with a detailed step-by-step solution. This problem covers algebraic concepts such as factoring, difference of squares, and sign analysis for polynomial inequalities. Suitable for students in Grades 10-12.

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