Let's solve the equation step by step:

Given:
\[
x + 3(x - 1) = 6 - 4(2x + 3)
\]

### Step 1: Distribute the constants across the parentheses.

For the left-hand side:
\[
3(x - 1) = 3x - 3
\]
So, the left-hand side becomes:
\[
x + 3x - 3 = 4x - 3
\]

For the right-hand side:
\[
-4(2x + 3) = -8x - 12
\]
So, the right-hand side becomes:
\[
6 - 8x - 12 = -8x - 6
\]

### Step 2: Rewrite the equation with the simplified expressions.
\[
4x - 3 = -8x - 6
\]

### Step 3: Move all the \(x\) terms to one side of the equation.
Add \(8x\) to both sides:
\[
4x + 8x - 3 = -6
\]
Simplifying, we get:
\[
12x - 3 = -6
\]

### Step 4: Move the constant term to the other side.
Add \(3\) to both sides:
\[
12x = -3
\]

### Step 5: Solve for \(x\) by dividing by 12.
\[
x = \frac{-3}{12} = \frac{-1}{4}
\]

So, the solution is:
\[
\boxed{x = -\frac{1}{4}}
\]

Would you like more details or have any questions about this solution?

Here are some related questions you might find interesting:
1. How do you solve equations with multiple variables?
2. What happens if there is no solution to an equation?
3. Can you explain how to factor quadratic equations?
4. How do you solve an equation with absolute values?
5. How would you graph the equation \(y = 4x - 3\)?

**Tip:** Always double-check your work by substituting the solution back into the original equation to ensure it satisfies both sides.

Learn how to solve the equation x + 3(x - 1) = 6 - 4(2x + 3) with a detailed step-by-step solution. This problem involves equations and the distributive property, suitable for students in Grades 7-9.

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