Let's analyze the given expression and options. We are asked to find all expressions equivalent to:

$\frac{-6x - 4}{2}$

Step 1: Simplify the given expression

We can simplify the expression by dividing both terms in the numerator by 2:

$\frac{-6x - 4}{2} = \frac{-6x}{2} + \frac{-4}{2} = -3x - 2$

So the simplified expression is $-3x - 2$.

Step 2: Analyze the options

• Option A: $-3x - 2$
  This is exactly the simplified form, so Option A is correct.

• Option B: $3x + 2$
  This is not equivalent to $-3x - 2$, so Option B is incorrect.

• Option C: $\frac{6x + 4}{2}$
  Simplifying this expression gives: $\frac{6x + 4}{2} = 3x + 2$ This is equivalent to Option B, which is incorrect, so Option C is also incorrect.

• Option D: $\frac{1}{2}(6x + 4)$
  Simplifying this expression: $\frac{1}{2}(6x + 4) = 3x + 2$ This is equivalent to $3x + 2$, so Option D is incorrect.

• Option E: $\frac{1}{2}(-3x - 2)$
  Simplifying this expression: $\frac{1}{2}(-3x - 2) = -\frac{3x}{2} - 1$ This is not equivalent to $-3x - 2$, so Option E is incorrect.

Conclusion:

The only correct answer is Option A.

Would you like more details on any of the steps? Here are some follow-up questions:

1. How do you distribute a fraction over an expression in parentheses?
2. Why isn't $\frac{1}{2}(-3x - 2)$ equivalent to $-3x - 2$?
3. How can you check if two expressions are equivalent?
4. What other methods can be used to simplify algebraic fractions?
5. How does factoring affect the equivalence of algebraic expressions?

Tip: Always simplify both the numerator and denominator separately before dividing when simplifying fractions.