Which of the following is equivalent to `(4x ^(2) -6x)/(2x +2)` ?

Options

A. `2x - (10)/(2x +2)`
B. `2x -5 + (10)/(2x +2)`
C. `2x -3`
D. `2x +5 - (10)/(2x +2)`

Answer

To solve the expression \((4x^2 - 6x) / (2x + 2)\), we will simplify the numerator and see if we can factor it in a way that relates to the denominator.

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ \frac{4x^2 - 6x}{2x + 2} \]

2. **Factor the numerator**:
We can factor out a common term from the numerator. Notice that both terms in the numerator have \(2x\) as a common factor:
\[ 4x^2 - 6x = 2x(2x - 3) \]
So, we rewrite the expression as:
\[ \frac{2x(2x - 3)}{2x + 2} \]

3. **Factor the denominator**:
The denominator can also be factored. We can factor out a 2:
\[ 2x + 2 = 2(x + 1) \]
Now, we can rewrite the expression:
\[ \frac{2x(2x - 3)}{2(x + 1)} \]

4. **Cancel the common factor**:
We can now cancel the common factor of 2 in the numerator and denominator:
\[ \frac{x(2x - 3)}{x + 1} \]

5. **Final expression**:
The simplified expression is:
\[ \frac{x(2x - 3)}{x + 1} \]

### Final Answer:
\[ \frac{x(2x - 3)}{x + 1} \]