The problem asks to subtract the two rational expressions:

$\frac{3x}{x+5} - \frac{3x - 6}{2x + 10}$

Step 1: Simplify the second denominator.

Notice that $2x + 10$ can be factored: $2x + 10 = 2(x + 5)$

Thus, the expression becomes: $\frac{3x}{x + 5} - \frac{3x - 6}{2(x + 5)}$

Step 2: Find a common denominator.

The common denominator between $x + 5$ and $2(x + 5)$ is $2(x + 5)$. We need to adjust the first fraction by multiplying both its numerator and denominator by 2: $\frac{3x}{x + 5} = \frac{2(3x)}{2(x + 5)} = \frac{6x}{2(x + 5)}$

Now the expression is: $\frac{6x}{2(x + 5)} - \frac{3x - 6}{2(x + 5)}$

Step 3: Subtract the numerators.

Now that the denominators are the same, we can subtract the numerators: $\frac{6x - (3x - 6)}{2(x + 5)} = \frac{6x - 3x + 6}{2(x + 5)} = \frac{3x + 6}{2(x + 5)}$

Step 4: Simplify the numerator.

The numerator $3x + 6$ can be factored: $3x + 6 = 3(x + 2)$

Thus, the expression becomes: $\frac{3(x + 2)}{2(x + 5)}$

This is the simplified form of the expression.

Would you like more details or further clarification?

Related Questions:

1. How do you find a common denominator for rational expressions?
2. What steps are involved in factoring polynomials?
3. Can you subtract fractions with different denominators without finding a common denominator?
4. How do you simplify a rational expression?
5. What are the general strategies for solving problems with rational expressions?

Tip:

Always look for opportunities to factor expressions to simplify rational functions, as this often makes common denominators and cancellation easier.