To solve the subtraction problem:

$\frac{3x}{x+2} - \frac{4x}{x-5}$

and simplify the expression, follow these steps:

Step 1: Find the common denominator

The denominators are $x+2$ and $x-5$. The least common denominator (LCD) is the product of these two terms: $(x+2)(x-5)$

Step 2: Rewrite each fraction with the LCD as the denominator

For $\frac{3x}{x+2}$, multiply numerator and denominator by $x-5$: $\frac{3x}{x+2} = \frac{3x(x-5)}{(x+2)(x-5)}$

For $\frac{4x}{x-5}$, multiply numerator and denominator by $x+2$: $\frac{4x}{x-5} = \frac{4x(x+2)}{(x+2)(x-5)}$

Step 3: Perform the subtraction

Now that the denominators are the same, subtract the numerators: $\frac{3x(x-5)}{(x+2)(x-5)} - \frac{4x(x+2)}{(x+2)(x-5)} = \frac{3x(x-5) - 4x(x+2)}{(x+2)(x-5)}$

Expand the terms in the numerator: $3x(x-5) = 3x^2 - 15x$ $4x(x+2) = 4x^2 + 8x$

Subtract the expanded terms: $(3x^2 - 15x) - (4x^2 + 8x) = 3x^2 - 15x - 4x^2 - 8x = -x^2 - 23x$

So the numerator becomes: $-x^2 - 23x$

Step 4: Write the simplified fraction

$\frac{-x^2 - 23x}{(x+2)(x-5)}$

Step 5: Factor if possible

Factor the numerator, $-x^2 - 23x$: $-x^2 - 23x = -x(x + 23)$

Thus, the simplified fraction is: $\frac{-x(x+23)}{(x+2)(x-5)}$

Final Answer:

$\boxed{\frac{-x(x+23)}{(x+2)(x-5)}}$

Would you like a more detailed explanation of any step? 😊

Here are 5 related questions to extend your understanding:

1. How do you find the least common denominator (LCD) for two fractions?
2. Why is it necessary to factor the numerator when simplifying fractions?
3. What are restrictions on $x$ in this problem (values that make the denominator zero)?
4. How can you check if your simplified answer is correct?
5. Can this expression be used for evaluating specific values of $x$? If so, how?

Tip: Always double-check for restrictions on variables when working with rational expressions to avoid undefined values.