We are asked to divide and simplify the expression:

$\frac{\frac{4x}{5}}{\frac{16x + 32}{5x + 10}}$

Step 1: Division of Fractions

When dividing fractions, we multiply the first fraction by the reciprocal of the second. So, the division becomes:

$\frac{4x}{5} \times \frac{5x + 10}{16x + 32}$

Step 2: Factorization

Next, let's factor the expressions where possible:

• $5x + 10$ can be factored as $5(x + 2)$.
• $16x + 32$ can be factored as $16(x + 2)$.

Now the expression becomes:

$\frac{4x}{5} \times \frac{5(x + 2)}{16(x + 2)}$

Step 3: Cancel Common Factors

We notice that $(x + 2)$ appears in both the numerator and denominator, so we can cancel this factor out. This leaves us with:

$\frac{4x}{5} \times \frac{5}{16}$

Step 4: Simplification

Now, multiply the numerators and denominators:

$\frac{4x \times 5}{5 \times 16} = \frac{20x}{80}$

Finally, simplify the fraction:

$\frac{20x}{80} = \frac{x}{4}$

Final Answer:

The simplified result is:

$\frac{x}{4}$

Would you like any further explanations or details about the process?

Here are some follow-up questions:

1. What happens if the variable $x$ equals $-2$?
2. How can you apply this method to more complex algebraic fractions?
3. How do you simplify a fraction when both the numerator and denominator contain polynomials?
4. Why do we cancel out the common factor $(x + 2)$ in the simplification step?
5. What would change if the original expression involved negative signs?

Tip: Always check for common factors in both the numerator and denominator before canceling them out—it helps simplify your work!