If x is not equal to 2 or -2 . Which s equivalent to `(3x^2 - 8x +4)/(x^2 - 4)` ?

Options

A. 3-8x
B. `(3x-2)/(x+2)`
C. `(3x-2)/(x-2)`
D. `(3x+2)/(x+2)`

Answer

To solve the problem, we need to simplify the expression \((3x^2 - 8x + 4)/(x^2 - 4)\).

### Step 1: Factor the numerator
We start with the numerator \(3x^2 - 8x + 4\). We can factor this quadratic expression.

1. We can rewrite \(3x^2 - 8x + 4\) as:
\[ 3x^2 - 6x - 2x + 4 \]
2. Now, we can group the terms:
\[ (3x^2 - 6x) + (-2x + 4) \]
3. Factor out the common factors from each group:
\[ 3x(x - 2) - 2(x - 2) \]
4. Now, we can factor out \((x - 2)\):
\[ (3x - 2)(x - 2) \]

### Step 2: Factor the denominator
Next, we factor the denominator \(x^2 - 4\):
1. Recognizing that \(x^2 - 4\) is a difference of squares, we can factor it as:
\[ (x - 2)(x + 2) \]

### Step 3: Rewrite the expression
Now we can rewrite the entire expression:
\[ \frac{(3x - 2)(x - 2)}{(x - 2)(x + 2)} \]

### Step 4: Cancel common factors
Since \(x \neq 2\), we can safely cancel the \((x - 2)\) terms from the numerator and the denominator:
\[ \frac{3x - 2}{x + 2} \]

### Conclusion
Thus, the expression \(\frac{3x^2 - 8x + 4}{x^2 - 4}\) simplifies to:
\[ \frac{3x - 2}{x + 2} \]

### Final Answer
The equivalent expression is \(\frac{3x - 2}{x + 2}\).

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