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The expression to simplify is:

\[
\frac{7x^3 - 56x^2}{x^2 + x - 72}
\]

Let's work through the steps to simplify this expression and determine any restrictions on the variable \(x\).

### Step 1: Factor both the numerator and the denominator

#### Numerator: \( 7x^3 - 56x^2 \)
- Factor out the common term \(7x^2\):

\[
7x^2(x - 8)
\]

So, the numerator becomes \( 7x^2(x - 8) \).

#### Denominator: \( x^2 + x - 72 \)
- We need to factor \( x^2 + x - 72 \). We look for two numbers whose product is \(-72\) and whose sum is \(1\).
  These numbers are \(9\) and \(-8\), so the denominator factors as:

\[
(x + 9)(x - 8)
\]

### Step 2: Simplify the expression

The expression now looks like:

\[
\frac{7x^2(x - 8)}{(x + 9)(x - 8)}
\]

We can cancel out the common factor \(x - 8\), but we must note that \(x \neq 8\) to avoid division by zero.

So, the simplified expression becomes:

\[
\frac{7x^2}{x + 9}
\]

### Step 3: State the restrictions on the variable

From the denominator of the original expression, we know that the values of \(x\) that would make the denominator zero are \(x = -9\) and \(x = 8\). Therefore, \(x\) cannot be \(-9\) or \(8\).

### Final Answer:
The simplified expression is:

\[
\frac{7x^2}{x + 9}
\]

with restrictions:

\[
x \neq -9, \quad x \neq 8
\]

---

Would you like more details on any part of the process, or do you have any other questions? 

Here are five related questions you can explore:
1. How would you solve a similar rational expression with more complex factors?
2. What are other methods to determine restrictions on variables in rational expressions?
3. How can you check if a simplification of a rational expression is correct?
4. How do you graph a rational function, like \(\frac{7x^2}{x+9}\), and interpret its behavior near restricted points?
5. What is the difference between simplifying and solving a rational equation?

**Tip:** Always check the original denominator before canceling terms to ensure you're aware of any restrictions on the variable.

Simplify and state any restrictions on the variables: (7x^3 - 56x^2) / (x^2 + x - 72)

Learn how to simplify the rational expression (7x^3 - 56x^2) / (x^2 + x - 72) with step-by-step instructions. Understand how to factor polynomials, simplify expressions, and identify restrictions on variables. This solution is suitable for students in Grades 9-10.

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