If the expression `(3x^2)/(x+2)` is written in the equivalent form `12/(x+2)+A` , what is A in terms of x ?

Options

A. 3x-6
B. 3x+6
C. `3x^2`
D. `3x^2-1`

Answer

To find the value of \( A \) in the expression \( \frac{3x^2}{x+2} = \frac{12}{x+2} + A \), we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ \frac{3x^2}{x+2} = \frac{12}{x+2} + A \]

### Step 2: Isolate \( A \)
To isolate \( A \), we can rearrange the equation:
\[ A = \frac{3x^2}{x+2} - \frac{12}{x+2} \]

### Step 3: Combine the fractions
Since both terms on the right side have the same denominator, we can combine them:
\[ A = \frac{3x^2 - 12}{x+2} \]

### Step 4: Factor the numerator
Next, we can factor the numerator \( 3x^2 - 12 \):
\[ 3x^2 - 12 = 3(x^2 - 4) \]
Recognizing that \( x^2 - 4 \) is a difference of squares, we can factor it further:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, we have:
\[ A = \frac{3(x - 2)(x + 2)}{x + 2} \]

### Step 5: Simplify the expression
Now, we can simplify the expression by canceling \( x + 2 \) in the numerator and denominator (assuming \( x \neq -2 \)):
\[ A = 3(x - 2) \]

### Final Answer
So, the value of \( A \) in terms of \( x \) is:
\[ A = 3(x - 2) \]

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