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

### Step 1: Break down the expression
We can separate the expression by dividing each term in the numerator by the denominator:

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

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

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

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

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

So, the simplified expression is:

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

Would you like further explanation or have any questions?

Here are 5 related questions for further practice:
1. How would you simplify \(\frac{36x^4 + 48x^3 + 60x^2}{12x^3}\)?
2. What is the result of \(\frac{45x^5 + 30x^4 + 15x^3}{15x^3}\)?
3. How can you simplify \(\frac{56x^4 + 32x^2 + 8x}{8x^2}\)?
4. What happens if you divide \(\frac{28x^3 + 42x^2 + 14x}{14x}\)?
5. How do you factor and simplify \(\frac{18x^3 + 12x^2 + 6x}{6x^2}\)?

**Tip:** When simplifying rational expressions, always look for the greatest common factor (GCF) first to make the process easier.

Learn how to simplify the rational expression (24x^3 + 32x^2 + 40x) / 8x^2 step-by-step. This problem involves algebraic concepts related to rational expressions and simplification. Suitable for high school students and those preparing for advanced algebra.

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