The expression provided is:

$\frac{5x + 20}{15x} \div \frac{x^2 - 16}{3x - 12}$

Step 1: Simplify the first fraction

The numerator of the first fraction is $5x + 20$. This can be factored as: $5x + 20 = 5(x + 4)$ So the first fraction becomes: $\frac{5(x + 4)}{15x}$ Now, simplify this by canceling out the common factor of 5 from the numerator and denominator: $\frac{x + 4}{3x}$

Step 2: Simplify the second fraction

The numerator of the second fraction is $x^2 - 16$, which is a difference of squares: $x^2 - 16 = (x - 4)(x + 4)$ The denominator of the second fraction is $3x - 12$, which can be factored as: $3x - 12 = 3(x - 4)$ So the second fraction becomes: $\frac{(x - 4)(x + 4)}{3(x - 4)}$ Now, cancel out the common factor $(x - 4)$ from the numerator and denominator: $\frac{x + 4}{3}$

Step 3: Rewrite the division as multiplication

Now, the original division expression becomes: $\frac{x + 4}{3x} \times \frac{3}{x + 4}$

Step 4: Simplify the expression

Cancel out the common factor $x + 4$ from the numerator and denominator: $\frac{1}{3x} \times 3 = \frac{1}{x}$

Final Answer:

The simplified form of the expression is: $\frac{1}{x}$

Would you like further details on any step or have any questions?

Here are some related questions you could explore:

1. How do you factor a quadratic expression like $x^2 - 16$?
2. Why is the division of fractions rewritten as multiplication?
3. How do you handle expressions that involve dividing polynomials?
4. What is the process for canceling terms in fractions?
5. How can you identify when a binomial is a difference of squares?

Tip: Always check for common factors in both the numerator and denominator to simplify fractions efficiently.