Select the correct choice below and fill in any answer boxes within your choice.
A.
StartFraction x squared minus 9 Over x squared EndFraction times StartFraction x squared minus 3 x Over x squared plus 7 x minus 30 EndFraction
equals
  
StartFraction x plus 3 Over x left parenthesis x plus 10 right parenthesis EndFraction
​, xnot equals
  
0 comma negative 10 comma 3

We are tasked with simplifying the following expression:

\[
\frac{x^2 - 9}{x^2} \times \frac{x^2 - 3x}{x^2 + 7x - 30}
\]

### Step 1: Factor the expressions
Let's begin by factoring the numerators and denominators where possible.

1. **Factor \(x^2 - 9\):**
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]

2. **Factor \(x^2 - 3x\):**
   Factor out an \(x\):
   \[
   x^2 - 3x = x(x - 3)
   \]

3. **Factor \(x^2 + 7x - 30\):**
   Find two numbers that multiply to \(-30\) and add up to \(7\). These numbers are \(10\) and \(-3\). Thus:
   \[
   x^2 + 7x - 30 = (x - 3)(x + 10)
   \]

### Step 2: Substitute the factored forms
Now substitute the factored expressions into the original equation:

\[
\frac{(x - 3)(x + 3)}{x^2} \times \frac{x(x - 3)}{(x - 3)(x + 10)}
\]

### Step 3: Simplify the expression
Cancel out the common terms:

- The \((x - 3)\) terms in the numerator and denominator cancel out.
- The \(x^2\) in the first denominator becomes \(x \times x\), and one \(x\) cancels with the \(x\) in the second numerator.

The simplified expression is now:

\[
\frac{x + 3}{x(x + 10)}
\]

### Step 4: Set the domain restrictions
The restrictions on \(x\) come from values that would make any denominator zero. These values are:

1. From \(x^2\): \(x \neq 0\)
2. From \(x - 3\): \(x \neq 3\)
3. From \(x + 10\): \(x \neq -10\)

Thus, the domain restrictions are \(x \neq 0\), \(x \neq 3\), and \(x \neq -10\).

### Final Answer:

\[
\frac{x + 3}{x(x + 10)}, \quad x \neq 0, -10, 3
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you factor quadratic expressions?
2. What are the steps to cancel out terms in a rational expression?
3. How can you determine domain restrictions from rational expressions?
4. What happens if a value that makes a denominator zero is included in the solution?
5. Can factoring help in simplifying complex algebraic fractions?

**Tip:** Always check for common factors in both the numerator and denominator to simplify expressions easily.

Select the correct choice below and fill in any answer boxes within your choice. A. StartFraction x squared minus 9 Over x squared EndFraction times StartFraction x squared minus 3 x Over x squared plus 7 x minus 30 EndFraction equals StartFraction x plus 3 Over x left parenthesis x plus 10 right parenthesis EndFraction, xnot equals 0 comma negative 10 comma 3

Learn how to simplify the rational expression (x^2 - 9) / x^2 * (x^2 - 3x) / (x^2 + 7x - 30) step-by-step using factoring techniques. This problem covers key algebraic concepts such as factoring difference of squares, simplifying rational expressions, and determining domain restrictions. Suitable for students in Grades 9-10.

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