The given equation is:

$\frac{-4x}{x+1} = -1 + \frac{x}{x+1}$

Let's solve this step by step.

Step 1: Eliminate the fractions

To eliminate the denominators, multiply both sides of the equation by $x + 1$:

$(x + 1) \times \frac{-4x}{x+1} = (x + 1) \times \left( -1 + \frac{x}{x+1} \right)$

This simplifies to:

$-4x = (x + 1)(-1) + x$

Step 2: Simplify the right-hand side

Distribute $-1$ on the right-hand side:

$-4x = - (x + 1) + x$

This becomes:

$-4x = -x - 1 + x$

Simplify:

$-4x = -1$

Step 3: Solve for $x$

Now, divide both sides by $-4$:

$x = \frac{1}{4}$

Thus, the solution is:

$x = \frac{1}{4}$

Would you like further details or have any questions?

Here are 5 related questions to extend the concept:

1. How would you check if $x = \frac{1}{4}$ is indeed a solution to the original equation?
2. What happens if $x = -1$? Why must $x \neq -1$ in this equation?
3. How can you apply the same steps to solve equations with different rational expressions?
4. How can factoring be used as an alternative method in solving equations like this one?
5. What would the graph of both sides of the equation look like, and how would they intersect?

Tip: Always check the solution by substituting it back into the original equation, especially in rational expressions where the denominator may introduce restrictions.