The expression shown asks to subtract two rational functions:

$\frac{x - 6}{x + 2} - \frac{2}{x + 5}$

Step-by-step solution:

1. Find the common denominator: The denominators are $x + 2$ and $x + 5$, so the common denominator is $(x + 2)(x + 5)$.

2. Rewrite the terms with the common denominator: $\frac{x - 6}{x + 2} = \frac{(x - 6)(x + 5)}{(x + 2)(x + 5)}$ $\frac{2}{x + 5} = \frac{2(x + 2)}{(x + 2)(x + 5)}$

3. Subtract the numerators: Now, subtract the numerators: $\frac{(x - 6)(x + 5) - 2(x + 2)}{(x + 2)(x + 5)}$

4. Expand the numerators:

   • Expand $(x - 6)(x + 5)$: $(x - 6)(x + 5) = x^2 + 5x - 6x - 30 = x^2 - x - 30$
   • Expand $2(x + 2)$: $2(x + 2) = 2x + 4$

5. Combine the results: $\frac{x^2 - x - 30 - (2x + 4)}{(x + 2)(x + 5)} = \frac{x^2 - x - 30 - 2x - 4}{(x + 2)(x + 5)}$

6. Simplify the numerator: Combine like terms: $x^2 - x - 2x - 30 - 4 = x^2 - 3x - 34$

   So, the simplified expression is: $\frac{x^2 - 3x - 34}{(x + 2)(x + 5)}$

Thus, the final simplified expression is:

$\frac{x^2 - 3x - 34}{(x + 2)(x + 5)}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you multiply binomials such as $(x - 6)(x + 5)$?
2. Can you simplify rational expressions in different ways?
3. How would you solve rational equations involving subtraction?
4. What are common mistakes when subtracting rational expressions?
5. How do you factor polynomials like $x^2 - 3x - 34$ if needed?

Tip: Always check if the numerator can be factored further before concluding that your expression is fully simplified.