To solve the problem, we need to analyze the quadratic expression given:
\[ x^2 + kx + 12 \]
where \( k \) is a negative integer. We need to find a possible value of \( k \).
### Step 1: Identify the Product and Sum of Roots
In a quadratic equation of the form \( ax^2 + bx + c \), the product of the roots (let's call them \( \alpha \) and \( \beta \)) is given by:
\[ \alpha \beta = \frac{c}{a} \]
And the sum of the roots is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
For our equation \( x^2 + kx + 12 \):
- Here, \( a = 1 \), \( b = k \), and \( c = 12 \).
- Therefore, the product of the roots is:
\[ \alpha \beta = 12 \]
- And the sum of the roots is:
\[ \alpha + \beta = -k \]
### Step 2: Factor Pairs of 12
Next, we need to find the pairs of integers that multiply to give 12. The possible pairs (considering both positive and negative integers) are:
1. \( 1 \times 12 \)
2. \( 2 \times 6 \)
3. \( 3 \times 4 \)
4. \( -1 \times -12 \)
5. \( -2 \times -6 \)
6. \( -3 \times -4 \)
### Step 3: Determine the Values of \( k \)
Since \( k \) is a negative integer, we need to consider the pairs where both factors are negative. This will ensure that their sum (which gives us \( -k \)) is a positive integer.
From the negative pairs:
- For \( -1 \) and \( -12 \):
- Sum: \( -1 + (-12) = -13 \) → \( k = 13 \) (not valid since \( k \) must be negative)
- For \( -2 \) and \( -6 \):
- Sum: \( -2 + (-6) = -8 \) → \( k = 8 \) (not valid since \( k \) must be negative)
- For \( -3 \) and \( -4 \):
- Sum: \( -3 + (-4) = -7 \) → \( k = 7 \) (not valid since \( k \) must be negative)
### Step 4: Valid Negative Integer Values for \( k \)
Now, we need to find the valid negative integer values for \( k \) from the positive pairs:
- From the pairs \( (1, 12) \), \( (2, 6) \), and \( (3, 4) \):
- The sums are \( 13, 8, \) and \( 7 \) respectively, which are all positive.
Thus, the only valid negative integer values for \( k \) that we can consider are:
- \( k = -1, -2, -3, -4, -6, -12 \)
### Step 5: Check the Options
Now, we check the given options for possible values of \( k \):
- If the options include \( -13, -8, -7 \) etc., we can see that:
- \( -13 \) is valid.
- \( -8 \) is not valid.
- \( -7 \) is not valid.
### Conclusion
Thus, the possible value of \( k \) is:
\[ \boxed{-13} \]
