`f(x)=(2-3x)(x+3)+4(x^(2)-6)`
What is the sum of the zeros of function f defined by the equation above?

Options

A. `3`
B. `6`
C. `7`
D. `11`

Answer

To find the sum of the zeros of the function \( f(x) = (2 - 3x)(x + 3) + 4(x^2 - 6) \), we will follow these steps:

### Step 1: Expand the function
We start by expanding the expression for \( f(x) \).

\[ f(x) = (2 - 3x)(x + 3) + 4(x^2 - 6) \]

First, we expand \( (2 - 3x)(x + 3) \):

\[ = 2x + 6 - 3x^2 - 9x \]

Now, we expand \( 4(x^2 - 6) \):

\[ = 4x^2 - 24 \]

Combining these results, we have:

\[ f(x) = (2x + 6 - 3x^2 - 9x) + (4x^2 - 24) \]

### Step 2: Combine like terms
Now we combine like terms:

\[ f(x) = (-3x^2 + 4x^2) + (2x - 9x) + (6 - 24) \]

This simplifies to:

\[ f(x) = x^2 - 7x - 18 \]

### Step 3: Set the function equal to zero
To find the zeros of the function, we set \( f(x) = 0 \):

\[ x^2 - 7x - 18 = 0 \]

### Step 4: Factor the quadratic
Next, we factor the quadratic equation:

\[ x^2 - 7x - 18 = (x + 2)(x - 9) = 0 \]

### Step 5: Solve for the zeros
Setting each factor equal to zero gives us:

\[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]

### Step 6: Calculate the sum of the zeros
Now we calculate the sum of the zeros:

\[ -2 + 9 = 7 \]

Thus, the sum of the zeros of the function \( f(x) \) is \( 7 \).

### Final Answer
The sum of the zeros of the function \( f(x) \) is \( 7 \).

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