If the expression `(9x^(2))/(3x-2)` is rewritten in the forms `A=(4)/(3x-2)`, which of the following gives A in terms of x?

Options

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

Explanation

To solve the problem, we need to rewrite the expression \(\frac{9x^2}{3x-2}\) in the form \(A + \frac{4}{3x-2}\). Let's go through the steps to find \(A\) in terms of \(x\).

### Step-by-Step Solution

1. **Start with the given expression**:
\[ A + \frac{4}{3x-2} = \frac{9x^2}{3x-2} \]

2. **Isolate \(A\)**:
To isolate \(A\), we can subtract \(\frac{4}{3x-2}\) from both sides:
\[ A = \frac{9x^2}{3x-2} - \frac{4}{3x-2} \]

3. **Combine the fractions**:
Since both terms on the right side have the same denominator, we can combine them:
\[ A = \frac{9x^2 - 4}{3x-2} \]

4. **Recognize the numerator as a difference of squares**:
The expression \(9x^2 - 4\) can be factored using the difference of squares formula:
\[ 9x^2 - 4 = (3x)^2 - (2)^2 = (3x - 2)(3x + 2) \]

5. **Substitute the factored form back into the equation**:
Now we can substitute this back into our expression for \(A\):
\[ A = \frac{(3x - 2)(3x + 2)}{3x - 2} \]

6. **Simplify the expression**:
We can cancel out the common factor \((3x - 2)\) in the numerator and denominator (as long as \(3x - 2 \neq 0\)):
\[ A = 3x + 2 \]

### Final Answer
Thus, \(A\) in terms of \(x\) is:
\[ A = 3x + 2 \]