$$ \frac{5}{x+3} $$$$ + $$$$  $$3x/x+4 =2

The equation you've provided is:

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

To solve this equation, follow these steps:

### Step 1: Eliminate the denominators
The least common denominator (LCD) of $ x+3 $ and $ x+4 $ is $ (x+3)(x+4) $.

Multiply both sides of the equation by the LCD to get rid of the denominators:

\[
(x+3)(x+4) \left( \frac{5}{x+3} + \frac{3x}{x+4} \right) = 2(x+3)(x+4)
\]

### Step 2: Simplify the equation
Distribute the $ (x+3)(x+4) $ to each term:

\[
5(x+4) + 3x(x+3) = 2(x+3)(x+4)
\]

Now simplify both sides:

\[
5x + 20 + 3x^2 + 9x = 2(x^2 + 7x + 12)
\]

### Step 3: Expand and combine like terms
On the left-hand side:

\[
3x^2 + 14x + 20
\]

On the right-hand side, expand:

\[
2(x^2 + 7x + 12) = 2x^2 + 14x + 24
\]

### Step 4: Set the equation to zero
Move all terms to one side:

\[
3x^2 + 14x + 20 - (2x^2 + 14x + 24) = 0
\]

Simplify the equation:

\[
x^2 - 4 = 0
\]

### Step 5: Solve for $ x $
This is a simple quadratic equation:

\[
x^2 = 4
\]

Take the square root of both sides:

\[
x = \pm 2
\]

### Step 6: Check for extraneous solutions
We must ensure that the solutions do not make the original denominators zero. The denominators are $ x+3 $ and $ x+4 $.

- For $ x = 2 $, both denominators are non-zero.
- For $ x = -2 $, both denominators are non-zero.

Therefore, the solutions are:

\[
x = 2 \quad \text{or} \quad x = -2
\]

### Final Answer:
\[
x = 2 \quad \text{or} \quad x = -2
\]

Would you like more details on any step or have questions?

Here are some related questions to explore:

1. How do you find the least common denominator (LCD) in rational equations?
2. What are common methods for solving quadratic equations?
3. How can you check for extraneous solutions in rational equations?
4. What are other types of equations that might result in extraneous solutions?
5. How can multiplying both sides of an equation by the LCD simplify the solving process?

**Tip:** Always check for extraneous solutions when solving rational equations to ensure valid results.

Solve the equation: $ \frac{5}{x+3} + \frac{3x}{x+4} = 2 $

Learn how to solve the rational equation $ \frac{5}{x+3} + \frac{3x}{x+4} = 2 $ with a detailed step-by-step solution. This tutorial covers key concepts such as finding the least common denominator, simplifying rational equations, and solving quadratic equations. Suitable for students in Grades 9-12.

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