Calculate the value of 'K', if x = k is a solution of the quadratic polynomial `x^2 + 4x + 3`.

Options

A. 1
B. `-1 `
C. 3
D. `-4`

Answer

To find the value of 'K' such that \( x = k \) is a solution of the quadratic polynomial \( x^2 + 4x + 3 \), we need to set up the equation and solve for \( k \).

### Step-by-Step Solution:

1. **Set up the equation**: Since \( k \) is a solution of the polynomial, we can substitute \( k \) into the polynomial equation:
\[ k^2 + 4k + 3 = 0 \]

2. **Factor the quadratic**: We need to factor the quadratic expression \( k^2 + 4k + 3 \). We look for two numbers that add up to 4 (the coefficient of \( k \)) and multiply to 3 (the constant term). The numbers that satisfy this are 3 and 1.
\[ k^2 + 3k + 1k + 3 = 0 \]

3. **Group the terms**: We can group the terms to factor by grouping:
\[ (k^2 + 3k) + (1k + 3) = 0 \]

4. **Factor out common terms**: From the first group, we can factor out \( k \), and from the second group, we can factor out 1:
\[ k(k + 3) + 1(k + 3) = 0 \]

5. **Factor out the common binomial**: Now we can factor out \( (k + 3) \):
\[ (k + 3)(k + 1) = 0 \]

6. **Set each factor to zero**: To find the values of \( k \), we set each factor equal to zero:
\[ k + 3 = 0 \quad \text{or} \quad k + 1 = 0 \]

7. **Solve for \( k \)**:
- From \( k + 3 = 0 \):
\[ k = -3 \]
- From \( k + 1 = 0 \):
\[ k = -1 \]

Thus, the values of \( k \) are \( k = -3 \) and \( k = -1 \).

### Conclusion:
The possible values of \( k \) are \( -3 \) and \( -1 \). If the question specifies a particular option, then you would select the appropriate one based on the options given.