https://file.aiquickdraw.com/mathbot/file/1727252702555uo3timrr.jpg

Let's break down the problem from the image:

We are tasked with simplifying the following expression:

\[
\frac{3x}{8x + 48} + \frac{6}{x^2 + 12x + 36}
\]

### Step 1: Factoring the denominators
First, let's factor both denominators:

1. \( 8x + 48 \) can be factored as:
   \[
   8(x + 6)
   \]
   
2. \( x^2 + 12x + 36 \) is a quadratic expression that can be factored as:
   \[
   (x + 6)(x + 6) = (x + 6)^2
   \]

### Step 2: Rewrite the expression with factored denominators
Now, rewrite the entire expression with the factored denominators:
\[
\frac{3x}{8(x + 6)} + \frac{6}{(x + 6)^2}
\]

### Step 3: Find a common denominator
The least common denominator (LCD) between \( 8(x + 6) \) and \( (x + 6)^2 \) is:
\[
8(x + 6)^2
\]

### Step 4: Rewrite each fraction with the common denominator
1. For \( \frac{3x}{8(x + 6)} \), multiply both numerator and denominator by \( (x + 6) \):
   \[
   \frac{3x(x + 6)}{8(x + 6)^2} = \frac{3x^2 + 18x}{8(x + 6)^2}
   \]

2. The second fraction, \( \frac{6}{(x + 6)^2} \), already has the denominator \( (x + 6)^2 \), so multiply both numerator and denominator by 8:
   \[
   \frac{6 \times 8}{8(x + 6)^2} = \frac{48}{8(x + 6)^2}
   \]

### Step 5: Combine the fractions
Now that both fractions have the same denominator, we can combine them:
\[
\frac{3x^2 + 18x}{8(x + 6)^2} + \frac{48}{8(x + 6)^2} = \frac{3x^2 + 18x + 48}{8(x + 6)^2}
\]

### Step 6: Conclusion
The simplified expression is:
\[
\frac{3x^2 + 18x + 48}{8(x + 6)^2}
\]

Looking at the options provided in the image, this matches the first option: 
\[
\boxed{\frac{3x^2 + 18x + 48}{8(x + 6)(x + 6)}}
\]

Would you like further clarification on any part of the process? 

### Here are 5 related questions to extend the topic:
1. How do you solve expressions where the denominators do not factor easily?
2. Can you explain the steps to simplify more complex rational expressions?
3. How do you find the least common denominator for multiple fractions?
4. What methods can be used to factor quadratics with no obvious factorization?
5. How does simplifying rational expressions help solve equations?

### Tip:
When combining fractions with different denominators, always ensure you find the least common denominator, as it simplifies the process of adding or subtracting the fractions.

Simplify the expression: (3x / (8x + 48)) + (6 / (x^2 + 12x + 36)).

This tutorial simplifies the rational expression (3x / (8x + 48)) + (6 / (x^2 + 12x + 36)) by factoring the denominators and finding the least common denominator. This solution is suitable for students in grades 8-10 and covers factoring, rational expressions, and algebraic simplification.

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