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To solve the equation:

\[
\frac{x}{x+3} + \frac{1}{4} = \frac{x}{2x+6}
\]

### Step 1: Simplify the Right Side
Notice that \(2x + 6\) can be factored as:
\[
2x + 6 = 2(x + 3)
\]
Thus, the equation becomes:
\[
\frac{x}{x+3} + \frac{1}{4} = \frac{x}{2(x+3)}
\]

### Step 2: Eliminate Denominators
To eliminate the denominators, multiply each term by the least common multiple of the denominators, which is \(4 \cdot 2(x+3) = 8(x+3)\).

Multiplying through by \(8(x+3)\), we get:
\[
8(x+3) \cdot \frac{x}{x+3} + 8(x+3) \cdot \frac{1}{4} = 8(x+3) \cdot \frac{x}{2(x+3)}
\]

Simplifying each term:
1. For the first term: \(8(x+3) \cdot \frac{x}{x+3} = 8x\).
2. For the second term: \(8(x+3) \cdot \frac{1}{4} = 2(x+3) = 2x + 6\).
3. For the third term: \(8(x+3) \cdot \frac{x}{2(x+3)} = 4x\).

This gives us the equation:
\[
8x + 2x + 6 = 4x
\]

### Step 3: Simplify and Solve for \(x\)
Combine like terms on the left side:
\[
10x + 6 = 4x
\]
Subtract \(4x\) from both sides:
\[
6x + 6 = 0
\]
Subtract 6 from both sides:
\[
6x = -6
\]
Divide by 6:
\[
x = -1
\]

### Solution
The solution to the equation is:
\[
x = -1
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the least common multiple of denominators in an equation?
2. What is the method to factor polynomials like \(2x + 6\)?
3. How can you check if a solution satisfies the original equation?
4. What is the importance of clearing fractions in solving rational equations?
5. Are there other methods to solve rational equations without clearing fractions?

### Tip:
Always check your solution by substituting it back into the original equation to verify that it holds true.

Solve the equation \( \frac{x}{x+3} + \frac{1}{4} = \frac{x}{2x+6} \).

Learn how to solve the rational equation \( \frac{x}{x+3} + \frac{1}{4} = \frac{x}{2x+6} \) with a detailed step-by-step solution. This problem includes algebraic concepts like clearing fractions and factoring. Suitable for high school students in Grades 10-12.

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