If `( x - 1) and (x + 3)` are the factors of `x^(2) + k_(1) x + k_(2)` then

Options

A. `k_(1) = -2, k_(2) = - 3`
B. ` k_(1) = 2, k_(2) = - 3 `
C. ` k_(1) = 2, k_(2) = 3 `
D. ` k_(1) = - 2, k_(2) = 3`

Answer

To solve the problem, we need to find the values of \( k_1 \) and \( k_2 \) given that \( (x - 1) \) and \( (x + 3) \) are factors of the quadratic equation \( x^2 + k_1 x + k_2 \).

### Step-by-step Solution:

1. **Set up the equations using the factors:**
Since \( (x - 1) \) is a factor, substituting \( x = 1 \) into the equation should yield 0:
\[ 1^2 + k_1(1) + k_2 = 0 \]
Simplifying this gives:
\[ 1 + k_1 + k_2 = 0 \quad \text{(Equation 1)} \]

2. **Substitute the second factor:**
Since \( (x + 3) \) is also a factor, substituting \( x = -3 \) into the equation should also yield 0:
\[ (-3)^2 + k_1(-3) + k_2 = 0 \]
Simplifying this gives:
\[ 9 - 3k_1 + k_2 = 0 \quad \text{(Equation 2)} \]

3. **Rearranging the equations:**
From Equation 1, we can express \( k_2 \) in terms of \( k_1 \):
\[ k_2 = -1 - k_1 \]
Substitute this expression for \( k_2 \) into Equation 2:
\[ 9 - 3k_1 + (-1 - k_1) = 0 \]
Simplifying this gives:
\[ 9 - 3k_1 - 1 - k_1 = 0 \]
\[ 8 - 4k_1 = 0 \]

4. **Solving for \( k_1 \):**
Rearranging the equation gives:
\[ 4k_1 = 8 \]
Dividing both sides by 4:
\[ k_1 = 2 \]

5. **Finding \( k_2 \):**
Now substitute \( k_1 = 2 \) back into the expression for \( k_2 \):
\[ k_2 = -1 - k_1 = -1 - 2 = -3 \]

6. **Final values:**
Thus, we have:
\[ k_1 = 2 \quad \text{and} \quad k_2 = -3 \]

### Summary:
The values of \( k_1 \) and \( k_2 \) are:
\[ k_1 = 2, \quad k_2 = -3 \]