Let's simplify the expression \(\frac{24x^3 + 32x^2 + 40x}{8x^3}\).

### Step 1: Break down the expression
The expression can be rewritten by separating each term in the numerator:

\[
\frac{24x^3}{8x^3} + \frac{32x^2}{8x^3} + \frac{40x}{8x^3}
\]

### Step 2: Simplify each term individually
1. \(\frac{24x^3}{8x^3}\):
   \[
   \frac{24}{8} \times \frac{x^3}{x^3} = 3 \times 1 = 3
   \]

2. \(\frac{32x^2}{8x^3}\):
   \[
   \frac{32}{8} \times \frac{x^2}{x^3} = 4 \times \frac{1}{x} = \frac{4}{x}
   \]

3. \(\frac{40x}{8x^3}\):
   \[
   \frac{40}{8} \times \frac{x}{x^3} = 5 \times \frac{1}{x^2} = \frac{5}{x^2}
   \]

### Step 3: Combine the simplified terms
\[
3 + \frac{4}{x} + \frac{5}{x^2}
\]

So, the simplified expression is:

\[
\boxed{3 + \frac{4}{x} + \frac{5}{x^2}}
\]

Would you like more details on any part of this solution? 

Here are 5 related questions to deepen your understanding:
1. How would you simplify \(\frac{18x^4 + 27x^3 + 36x^2}{9x^2}\)?
2. What is the result of \(\frac{50x^5 + 25x^3 + 10x}{5x}\)?
3. How can you simplify \(\frac{40x^3 + 24x^2 + 8x}{8x}\)?
4. What happens if you divide \(\frac{15x^3 + 30x^2 + 45x}{15x^2}\)?
5. How do you factor and simplify \(\frac{64x^4 + 32x^2}{16x^2}\)?

**Tip:** When simplifying expressions, it's often useful to factor out common terms before dividing, as this can simplify the process considerably.

Learn how to simplify the rational expression (24x^3 + 32x^2 + 40x) / 8x^3 with a step-by-step solution. This problem involves algebraic simplification of rational expressions, suitable for students in Grades 7-9.

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