We are tasked with solving for $x$ in the equation:

$-2(2x + 3) + 1 = -x - 13 + 3x$

Step 1: Distribute the $-2$ on the left-hand side.

Distribute the $-2$ across the terms inside the parentheses:

$-2(2x) - 2(3) + 1 = -x - 13 + 3x$

This simplifies to:

$-4x - 6 + 1 = -x - 13 + 3x$

Step 2: Combine like terms on both sides.

On the left-hand side, combine $-6 + 1$:

$-4x - 5 = -x - 13 + 3x$

On the right-hand side, combine the $x$-terms $-x + 3x$:

$-4x - 5 = 2x - 13$

Step 3: Move all terms involving $x$ to one side.

Add $4x$ to both sides to eliminate the $-4x$ on the left-hand side:

$-4x + 4x - 5 = 2x + 4x - 13$

This simplifies to:

$-5 = 6x - 13$

Step 4: Isolate the $x$-term.

Add 13 to both sides to move the constant term to the left-hand side:

$-5 + 13 = 6x$

This simplifies to:

$8 = 6x$

Step 5: Solve for $x$.

Divide both sides by 6 to isolate $x$:

$x = \frac{8}{6} = \frac{4}{3}$

Thus, the solution is:

$x = \frac{4}{3}$

Final Answer:

$x = \frac{4}{3}$

Would you like more details or further clarifications?

Here are some related questions:

1. How would the process change if the equation had fractional coefficients?
2. How do we handle distributive property with more complicated expressions?
3. What happens if there are absolute values in the equation?
4. How can we check if the solution is correct?
5. What are the steps to solve a quadratic equation?

Tip: Always double-check your work by substituting the solution back into the original equation.