The image contains five division problems involving rational expressions. Let’s solve them one by one.

1.

$\frac{\frac{x-4}{7x^3 + 42x^2}}{\frac{1}{7x^2}}$

To divide these fractions, we multiply by the reciprocal of the second fraction: $= \frac{x-4}{7x^3 + 42x^2} \cdot \frac{7x^2}{1}$

Simplify: $= \frac{(x-4) \cdot 7x^2}{7x^3 + 42x^2}$

Factor the denominator: $7x^3 + 42x^2 = 7x^2(x + 6)$

So, the expression becomes: $= \frac{7x^2(x-4)}{7x^2(x + 6)}$

Cancel out $7x^2$: $= \frac{x-4}{x+6}$

2.

$\frac{\frac{7}{2x^3 + 16x^2}}{\frac{1}{x+8}}$

Multiply by the reciprocal of the second fraction: $= \frac{7}{2x^3 + 16x^2} \cdot \frac{x+8}{1}$

Simplify: $= \frac{7(x+8)}{2x^3 + 16x^2}$

Factor the denominator: $2x^3 + 16x^2 = 2x^2(x + 8)$

So the expression becomes: $= \frac{7(x+8)}{2x^2(x + 8)}$

Cancel out $x+8$: $= \frac{7}{2x^2}$

3.

$\frac{\frac{3x^2}{5x^2}}{\frac{x+4}{7x^2 + 28x}}$

Multiply by the reciprocal of the second fraction: $= \frac{3x^2}{5x^2} \cdot \frac{7x^2 + 28x}{x+4}$

Factor the numerator: $7x^2 + 28x = 7x(x+4)$

So the expression becomes: $= \frac{3x^2}{5x^2} \cdot \frac{7x(x+4)}{x+4}$

Cancel out $x+4$ and $x^2$: $= \frac{3}{5} \cdot 7x = \frac{21x}{5}$

4.

$\frac{\frac{8x+4}{4}}{\frac{4x+2}{7x}}$

Multiply by the reciprocal of the second fraction: $= \frac{8x+4}{4} \cdot \frac{7x}{4x+2}$

Factor both numerators and denominators: $8x + 4 = 4(2x+1), \quad 4x + 2 = 2(2x+1)$

The expression becomes: $= \frac{4(2x+1)}{4} \cdot \frac{7x}{2(2x+1)}$

Cancel out $2x+1$ and simplify: $= \frac{7x}{2}$

5.

$\frac{\frac{1}{x+1}}{\frac{x+8}{3x^3 + 24x^2}}$

Multiply by the reciprocal of the second fraction: $= \frac{1}{x+1} \cdot \frac{3x^3 + 24x^2}{x+8}$

Factor the numerator: $3x^3 + 24x^2 = 3x^2(x+8)$

So the expression becomes: $= \frac{3x^2(x+8)}{(x+1)(x+8)}$

Cancel out $x+8$: $= \frac{3x^2}{x+1}$

Would you like more details or have any questions about the steps?

Here are 5 related questions you could explore:

1. How do you handle complex rational expressions involving polynomials with higher degrees?
2. What happens if the factors in the numerator and denominator do not cancel out?
3. How can we simplify rational expressions with more complicated numerators or denominators?
4. What strategies can be used to identify common factors in polynomials?
5. How do division and multiplication of rational expressions relate to division of simple fractions?

Tip: Always check for common factors in both the numerator and denominator before simplifying!