The question shown in the image asks you to divide the following rational expressions and simplify the result:

$\frac{\frac{x^2 - 16}{x^2 + 7x + 14}}{\frac{x^2 - 4x}{7x + 14}}$

To solve this, follow these steps:

Step 1: Factor the expressions.

• The numerator of the first fraction is $x^2 - 16$, which is a difference of squares. It factors as: $x^2 - 16 = (x - 4)(x + 4)$
• The denominator of the first fraction is $x^2 + 7x + 14$. Factor this quadratic: $x^2 + 7x + 14 = (x + 2)(x + 7)$
• The numerator of the second fraction is $x^2 - 4x$. Factor out an $x$: $x^2 - 4x = x(x - 4)$
• The denominator of the second fraction is $7x + 14$, which can be factored as: $7x + 14 = 7(x + 2)$

Step 2: Rewrite the expression.

Now substitute the factored forms into the expression: $\frac{\frac{(x - 4)(x + 4)}{(x + 2)(x + 7)}}{\frac{x(x - 4)}{7(x + 2)}}$

Step 3: Division of fractions.

When dividing fractions, multiply by the reciprocal of the second fraction: $= \frac{(x - 4)(x + 4)}{(x + 2)(x + 7)} \times \frac{7(x + 2)}{x(x - 4)}$

Step 4: Cancel common factors.

• $x - 4$ cancels with $x - 4$.
• $x + 2$ cancels with $x + 2$.

The simplified expression becomes: $\frac{(x + 4) \cdot 7}{(x + 7) \cdot x} = \frac{7(x + 4)}{x(x + 7)}$

Final Answer:

The simplified form of the expression is: $\frac{7(x + 4)}{x(x + 7)}$

Let me know if you would like further clarification or details!

Here are 5 related questions to expand on this topic:

1. How do you factor more complex quadratic expressions?
2. What are the key steps in simplifying rational expressions?
3. How do you handle operations with rational expressions when multiple variables are involved?
4. What are the rules for dividing fractions?
5. How can you determine the domain of a rational expression?

Tip: When simplifying rational expressions, always look for opportunities to factor and cancel common terms before multiplying or dividing.