The expression you've written is:

\[
\frac{8x}{x+4} + \frac{x+4}{12x^2}
\]

To simplify this, we can follow these steps:

### Step 1: Find a common denominator
The denominators in the expression are \( x+4 \) and \( 12x^2 \). The least common denominator (LCD) between these is \( 12x^2(x+4) \).

### Step 2: Rewrite each fraction with the LCD
Now, we rewrite each term so that both have the same denominator:

1. \(\frac{8x}{x+4}\) becomes:
\[
\frac{8x \cdot 12x^2}{(x+4) \cdot 12x^2} = \frac{96x^3}{12x^2(x+4)}
\]

2. \(\frac{x+4}{12x^2}\) remains as:
\[
\frac{x+4}{12x^2(x+4)}
\]

### Step 3: Combine the two fractions
Now that they have the same denominator, we can combine them:
\[
\frac{96x^3 + (x+4)}{12x^2(x+4)}
\]

Simplifying the numerator:
\[
96x^3 + x + 4
\]

Thus, the expression becomes:
\[
\frac{96x^3 + x + 4}{12x^2(x+4)}
\]

This is the simplified form of the original expression. If you need to factor further or analyze the behavior of this expression, you could do so depending on the context.

Would you like a detailed explanation of any specific part?

### Follow-up questions:
1. What are the critical values of \(x\) where the expression is undefined?
2. How does the expression behave as \(x\) approaches zero?
3. Can this expression be simplified further by factoring?
4. What is the value of this expression for a specific value of \(x\), like \(x = 2\)?
5. Can this expression be applied in a real-world context, such as in physics?

### Tip:
Always check for undefined points by examining the denominators for values of \(x\) that would make them zero.

Simplify the rational expression 8x/(x+4) + (x+4)/12x^2 step-by-step. Learn how to find a common denominator, combine terms, and simplify the result. Understand the key concepts involved in rational expressions and algebraic simplifications. Ideal for students in Grades 9-11.

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