The inequality provided is:

\[
- x^3 + 7x^2 + 9x > 63
\]

Let's solve this step by step.

### Step 1: Rearrange the inequality
Move all terms to one side of the inequality:

\[
- x^3 + 7x^2 + 9x - 63 > 0
\]

### Step 2: Analyze the cubic equation
To solve a cubic inequality like this one, we'll need to find the roots of the equation:

\[
- x^3 + 7x^2 + 9x - 63 = 0
\]

We'll solve this cubic equation first to find the critical points. After that, we can determine where the inequality is greater than zero by testing intervals between these points.

Let me calculate the roots of this cubic equation.The roots of the cubic equation \(- x^3 + 7x^2 + 9x - 63 = 0\) are:

\[
x = -3, \, x = 3, \, x = 7
\]

### Step 3: Determine intervals to test
The roots divide the real line into the following intervals:
1. \( (-\infty, -3) \)
2. \( (-3, 3) \)
3. \( (3, 7) \)
4. \( (7, \infty) \)

We will now test the sign of the cubic expression \( - x^3 + 7x^2 + 9x - 63 \) in each of these intervals to see where the inequality holds true.

Let's check each interval.

Learn how to solve the cubic inequality -x^3 + 7x^2 + 9x > 63 step by step. This problem covers key algebraic concepts such as solving cubic equations, finding polynomial roots, and analyzing intervals for inequalities. Suitable for high school students in Grades 9-12.

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