Let's simplify the expression $\frac{4x^2 - 12x + 24}{4x}$.

Step-by-Step Solution:

1. Distribute the denominator across the numerator: $\frac{4x^2}{4x} - \frac{12x}{4x} + \frac{24}{4x}$

2. Simplify each term individually:

   • The first term: $\frac{4x^2}{4x} = \frac{4 \cdot x^2}{4 \cdot x} = x$
   • The second term: $\frac{12x}{4x} = \frac{12 \cdot x}{4 \cdot x} = 3$
   • The third term: $\frac{24}{4x} = \frac{24}{4} \cdot \frac{1}{x} = 6 \cdot \frac{1}{x} = \frac{6}{x}$

3. Combine the simplified terms: $x - 3 + \frac{6}{x}$

So, the simplified expression is:

$x - 3 + \frac{6}{x}$

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

Related Questions:

1. How would the solution change if the expression was $4x^2 - 12x + 24x$?
2. What happens if you divide each term in $4x^2 - 12x + 24$ separately by $x$ instead of $4x$?
3. How can you factor the expression $4x^2 - 12x + 24$ before dividing by $4x$?
4. What if the expression was $4x^3 - 12x^2 + 24x$? How would you simplify it?
5. How would you graph the expression $x - 3 + \frac{6}{x}$?

Tip: When simplifying rational expressions, always check if the numerator can be factored to simplify the expression further before dividing.