`(2x+6)/(x^(2)+3x)-(x+3)/(x^(2)+x)`
Which of the following equivalent to the rational expression given above?

Options

A. `(-1)/(x)`
B. `(x+3)/(2x)`
C. `(x-1)/(x(x+1))`
D. `(x+9)/(2x^(2)+4x)`

Explanation

To solve the expression \((2x+6)/(x^{2}+3x)-(x+3)/(x^{2}+x)\) and find an equivalent rational expression, we will follow these steps:

### Step 1: Factor the Numerators and Denominators
1. **Numerator of the first fraction**: \(2x + 6\) can be factored as \(2(x + 3)\).
2. **Denominator of the first fraction**: \(x^2 + 3x\) can be factored as \(x(x + 3)\).
3. **Numerator of the second fraction**: \(x + 3\) is already factored.
4. **Denominator of the second fraction**: \(x^2 + x\) can be factored as \(x(x + 1)\).

So, we rewrite the expression:
\[ \frac{2(x + 3)}{x(x + 3)} - \frac{x + 3}{x(x + 1)} \]

### Step 2: Identify Common Terms
Notice that both fractions have a common term of \(x + 3\). We can factor this out:
\[ \frac{(x + 3)}{x} \left(2 - \frac{1}{x + 1}\right) \]

### Step 3: Find a Common Denominator
To combine the terms inside the parentheses, we need a common denominator:
- The common denominator for \(2\) and \(\frac{1}{x + 1}\) is \(x + 1\).

Rewriting \(2\) with the common denominator:
\[ 2 = \frac{2(x + 1)}{x + 1} \]

Now we can rewrite the expression:
\[ \frac{(x + 3)}{x} \left(\frac{2(x + 1) - 1}{x + 1}\right) \]

### Step 4: Simplify the Expression
Now, simplify the numerator:
\[ 2(x + 1) - 1 = 2x + 2 - 1 = 2x + 1 \]

So, we have:
\[ \frac{(x + 3)(2x + 1)}{x(x + 1)} \]

### Step 5: Final Expression
The final expression is:
\[ \frac{(x + 3)(2x + 1)}{x(x + 1)} \]

### Conclusion
This is the simplified form of the original expression.