Let's solve the equation step by step:

Given equation:
\[
3(x+2) - 4(x+6) = 3x + 8 + x
\]

### Step 1: Distribute the constants.
On the left-hand side:
\[
3(x+2) = 3x + 6
\]
\[
-4(x+6) = -4x - 24
\]
So, the left-hand side becomes:
\[
3x + 6 - 4x - 24
\]

On the right-hand side:
\[
3x + 8 + x = 4x + 8
\]

### Step 2: Combine like terms.
On the left-hand side:
\[
(3x - 4x) + (6 - 24) = -x - 18
\]

On the right-hand side:
\[
4x + 8
\]

Now, the equation is:
\[
-x - 18 = 4x + 8
\]

### Step 3: Solve for \(x\).
Add \(x\) to both sides:
\[
-18 = 5x + 8
\]

Subtract 8 from both sides:
\[
-18 - 8 = 5x
\]
\[
-26 = 5x
\]

Divide both sides by 5:
\[
x = \frac{-26}{5}
\]
\[
x = -5.2
\]

### Final Answer:
\[
x = -5.2
\]

Would you like further clarification or details on any of these steps?

Here are five related questions for deeper understanding:
1. What happens if you change the constants in the equation?
2. Can you solve a similar equation with different coefficients?
3. How do you handle equations with fractions?
4. What is the process for solving quadratic equations?
5. How can you verify the solution to this equation?

**Tip:** Always double-check your solution by substituting the value of \(x\) back into the original equation.

Learn how to solve the linear equation 3(x+2) - 4(x+6) = 3x + 8 + x with a detailed step-by-step breakdown. This problem covers key algebraic concepts like distribution, combining like terms, and solving for x. Ideal for students in Grades 8-10.

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