The problem asks to divide two rational expressions and select the correct simplification, with attention to any excluded values for $x$.

Step 1: Simplify the Expression

The given expression is:

$\frac{\frac{x^2 - 49}{x}}{\frac{x + 7}{x - 7}}$

This can be rewritten as a multiplication by taking the reciprocal of the denominator:

$\frac{x^2 - 49}{x} \times \frac{x - 7}{x + 7}$

Step 2: Factor the Numerator $x^2 - 49$

We recognize that $x^2 - 49$ is a difference of squares:

$x^2 - 49 = (x - 7)(x + 7)$

Step 3: Substitute the Factored Form

Substituting the factored form of $x^2 - 49$ into the expression:

$\frac{(x - 7)(x + 7)}{x} \times \frac{x - 7}{x + 7}$

Step 4: Cancel Common Factors

Now we can cancel out the common factors $(x + 7)$ from the numerator and denominator:

$\frac{(x - 7)}{x} \times (x - 7) = \frac{(x - 7)^2}{x}$

Step 5: Determine Excluded Values

The denominator of the original expression contains $x$ and $x - 7$, so we must exclude values of $x$ that make the denominator zero. These values are:

• $x \neq 0$ (from the term $\frac{1}{x}$)
• $x \neq 7$ (from the term $\frac{1}{x - 7}$)

Final Answer

The simplified expression is:

$\frac{(x - 7)^2}{x}, \quad x \neq 0, 7$

Correct Choice

The correct answer is Option A, with the conditions $x \neq 0$ and $x \neq 7$.

Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do you factor other quadratic expressions, such as $x^2 + bx + c$?
2. What are the steps to simplify more complex rational expressions?
3. How do you handle rational expressions with multiple variables?
4. What happens when you divide polynomials with more complicated denominators?
5. How do you find excluded values in rational expressions involving roots?

Tip: Always check for excluded values by setting each denominator in the expression equal to zero and solving for the variable.