The equation provided in the image is:

$\frac{1}{x^2 + 3x} - \frac{3}{x^2 + 2x - 3} = \frac{1}{x^2 - x}$

Step-by-step Solution:

1. Factor the Denominators: First, factor the quadratic expressions in the denominators:

   • $x^2 + 3x = x(x + 3)$
   • $x^2 + 2x - 3 = (x - 1)(x + 3)$
   • $x^2 - x = x(x - 1)$

2. Rewrite the Equation with Factored Denominators: The equation now becomes:

   $\frac{1}{x(x + 3)} - \frac{3}{(x - 1)(x + 3)} = \frac{1}{x(x - 1)}$

3. Find the Least Common Denominator (LCD): The least common denominator (LCD) of all terms is $x(x - 1)(x + 3)$. Multiply each term by the LCD:

   $\frac{1}{x(x + 3)} \cdot \frac{x(x - 1)(x + 3)}{x(x - 1)(x + 3)} - \frac{3}{(x - 1)(x + 3)} \cdot \frac{x(x - 1)(x + 3)}{x(x - 1)(x + 3)} = \frac{1}{x(x - 1)} \cdot \frac{x(x - 1)(x + 3)}{x(x - 1)(x + 3)}$

4. Simplify Each Term:

   • First term: $\frac{x(x - 1)(x + 3)}{x(x + 3)} = (x - 1)$
   • Second term: $\frac{3x(x - 1)(x + 3)}{(x - 1)(x + 3)} = 3x$
   • Third term: $\frac{x(x - 1)(x + 3)}{x(x - 1)} = (x + 3)$

   The equation becomes:

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

5. Solve for $x$: Simplify the left-hand side:

   $x - 1 - 3x = x + 3$

   Combine like terms:

   $-2x - 1 = x + 3$

   Move all terms involving $x$ to one side:

   $-2x - x = 3 + 1$

   Simplify:

   $-3x = 4$

   Solve for $x$:

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

Thus, the solution to the equation is $x = -\frac{4}{3}$.

Do you want more details or have any questions?

Here are 5 related questions to explore this topic further:

1. How do you factor quadratic expressions like $x^2 + 2x - 3$?
2. What is the method for finding the least common denominator in rational expressions?
3. How can you verify that $x = -\frac{4}{3}$ is the correct solution to the original equation?
4. What is the significance of excluding certain values from the domain of the expression (e.g., values that make the denominator zero)?
5. Can you simplify expressions with more complex denominators using partial fraction decomposition?

Tip: Always check for excluded values from the domain by identifying when the denominators are zero. In this case, exclude $x = 0$, $x = -3$, and $x = 1$.