If the function `f(x)=(3x^2+ax+a+3)/(x^2+x-2)` is continuous at `x=-2,` then the value of `f(-2)` is

Options

A. 0
B. `-1`
C. 1
D. 2

Explanation

To find the value of \( f(-2) \) for the function

\[ f(x) = \frac{3x^2 + ax + a + 3}{x^2 + x - 2} \]

given that it is continuous at \( x = -2 \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches \(-2\) is equal to \( f(-2) \).

### Step 1: Determine \( f(-2) \)

First, we need to evaluate the denominator at \( x = -2 \):

\[ x^2 + x - 2 = (-2)^2 + (-2) - 2 = 4 - 2 - 2 = 0 \]

Since the denominator is \( 0 \), we need to ensure that the numerator also equals \( 0 \) at \( x = -2 \) to avoid an undefined expression (0/0 form).

### Step 2: Evaluate the numerator at \( x = -2 \)

Now, we evaluate the numerator at \( x = -2 \):

\[ 3(-2)^2 + a(-2) + a + 3 = 3(4) - 2a + a + 3 = 12 - a + 3 = 15 - a \]

### Step 3: Set the numerator equal to zero

For the function to be continuous at \( x = -2 \), we set the numerator equal to zero:

\[ 15 - a = 0 \]

Solving for \( a \):

\[ a = 15 \]

### Step 4: Substitute \( a \) back into the function

Now, substituting \( a = 15 \) back into the function:

\[ f(x) = \frac{3x^2 + 15x + 15 + 3}{x^2 + x - 2} = \frac{3x^2 + 15x + 18}{x^2 + x - 2} \]

### Step 5: Factor the numerator and denominator

Next, we factor both the numerator and the denominator:

1. **Denominator**:

\[ x^2 + x - 2 = (x - 1)(x + 2) \]

2. **Numerator**:

To factor \( 3x^2 + 15x + 18 \), we can take out a common factor of 3:

\[ 3(x^2 + 5x + 6) = 3(x + 2)(x + 3) \]

### Step 6: Simplify the function

Now we can simplify \( f(x) \):

\[ f(x) = \frac{3(x + 2)(x + 3)}{(x - 1)(x + 2)} \]

We can cancel \( (x + 2) \) from the numerator and denominator (as long as \( x \neq -2 \)):

\[ f(x) = \frac{3(x + 3)}{x - 1} \quad \text{for } x \neq -2 \]

### Step 7: Find \( f(-2) \)

Now we can find \( f(-2) \) using the simplified function:

\[ f(-2) = \frac{3(-2 + 3)}{-2 - 1} = \frac{3(1)}{-3} = -1 \]

Thus, the value of \( f(-2) \) is

\[ \boxed{-1} \]