If `g(x)=2(x^(2)-6x-12)+3(k-x)` and g(x) is evenly divisible by x, then what is the value of k?

Options

A. `-8`
B. `-3`
C. `2`
D. `3`

Explanation

To solve the problem, we need to find the value of \( k \) such that the function \( g(x) \) is evenly divisible by \( x \). This means that when we substitute \( x = 0 \) into \( g(x) \), the result should be zero.

### Step 1: Write down the function
The given function is:
\[ g(x) = 2(x^2 - 6x - 12) + 3(k - x) \]

### Step 2: Expand the function
We will expand \( g(x) \) by distributing the constants:
\[ g(x) = 2(x^2) - 2(6x) - 2(12) + 3(k) - 3(x) \]
This simplifies to:
\[ g(x) = 2x^2 - 12x - 24 + 3k - 3x \]

### Step 3: Combine like terms
Now, we will combine the like terms:
\[ g(x) = 2x^2 + (-12x - 3x) + (3k - 24) \]
This simplifies to:
\[ g(x) = 2x^2 - 15x + (3k - 24) \]

### Step 4: Set the constant term to zero
For \( g(x) \) to be divisible by \( x \), the constant term must equal zero. The constant term in \( g(x) \) is \( 3k - 24 \). Therefore, we set it to zero:
\[ 3k - 24 = 0 \]

### Step 5: Solve for \( k \)
Now, we solve for \( k \):
\[ 3k = 24 \]
\[ k = \frac{24}{3} = 8 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{8} \]

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