If `cos^4 alpha +k and sin^4 alpha+k` are the roots of `x^2+lambda(2 x+1)=0 and sin^2 alpha+1 and cos^2 alpha +1` are the roots of `x^2 +8 x+4=0,` then the sum of the possible values of `lambda` is

Options

A. `2`
B. `-1`
C. `1`
D. `3`

Explanation

To solve the problem, we need to analyze the given equations and their roots step by step.

### Step 1: Identify the roots of the first equation
We are given that \( \cos^4 \alpha + k \) and \( \sin^4 \alpha + k \) are the roots of the equation:

\[ x^2 + \lambda(2x + 1) = 0 \]

This can be rewritten as:

\[ x^2 + 2\lambda x + \lambda = 0 \]

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( r_1 + r_2 = -\frac{b}{a} \) and the product of the roots \( r_1 r_2 = \frac{c}{a} \).

Here, the sum of the roots \( \cos^4 \alpha + k + \sin^4 \alpha + k \) is given by:

\[ \cos^4 \alpha + \sin^4 \alpha + 2k = -\frac{2\lambda}{1} = -2\lambda \]

### Step 3: Simplify the sum of the roots
Using the identity \( \cos^4 \alpha + \sin^4 \alpha = (\cos^2 \alpha + \sin^2 \alpha)^2 - 2\cos^2 \alpha \sin^2 \alpha \):

Since \( \cos^2 \alpha + \sin^2 \alpha = 1 \):

\[ \cos^4 \alpha + \sin^4 \alpha = 1 - 2\cos^2 \alpha \sin^2 \alpha = 1 - \frac{1}{2} \sin^2(2\alpha) \]

Thus, we can express the sum of the roots as:

\[ 1 - \frac{1}{2} \sin^2(2\alpha) + 2k = -2\lambda \]

### Step 4: Identify the roots of the second equation
We are also given that \( \sin^2 \alpha + 1 \) and \( \cos^2 \alpha + 1 \) are the roots of the equation:

\[ x^2 + 8x + 4 = 0 \]

Using Vieta's formulas again, we find:

\[ (\sin^2 \alpha + 1) + (\cos^2 \alpha + 1) = -8 \]

This simplifies to:

\[ \sin^2 \alpha + \cos^2 \alpha + 2 = -8 \implies 1 + 2 = -8 \implies 3 = -8 \text{ (not possible)} \]

### Step 5: Correct the approach
Let's correct the approach. The sum of the roots should be:

\[ \sin^2 \alpha + \cos^2 \alpha + 2 = -8 \implies 1 + 2 = -8 \implies 3 = -8 \text{ (impossible)} \]

This indicates a misunderstanding in the interpretation of the roots. We need to find the roots correctly.

### Step 6: Calculate the product of the roots
The product of the roots gives us:

\[ (\sin^2 \alpha + 1)(\cos^2 \alpha + 1) = 4 \]

Expanding this gives:

\[ \sin^2 \alpha \cos^2 \alpha + \sin^2 \alpha + \cos^2 \alpha + 1 = 4 \]

Using \( \sin^2 \alpha + \cos^2 \alpha = 1 \):

\[ \sin^2 \alpha \cos^2 \alpha + 1 + 1 = 4 \implies \sin^2 \alpha \cos^2 \alpha = 2 \]

### Step 7: Solve for \( \lambda \)
Now we go back to the equation for \( \lambda \):

From the earlier step, we have:

\[ 1 - \frac{1}{2} \sin^2(2\alpha) + 2k = -2\lambda \]

Substituting \( \sin^2(2\alpha) = 4\sin^2 \alpha \cos^2 \alpha = 8 \):

\[ 1 - \frac{1}{2}(8) + 2k = -2\lambda \implies 1 - 4 + 2k = -2\lambda \implies 2k - 3 = -2\lambda \]

### Step 8: Rearranging for \( \lambda \)
Rearranging gives:

\[ 2\lambda = 3 - 2k \implies \lambda = \frac{3 - 2k}{2} \]

### Step 9: Find possible values of \( \lambda \)
To find the sum of possible values of \( \lambda \), we need to consider the values of \( k \) that satisfy the conditions of the roots.

### Conclusion
The sum of the possible values of \( \lambda \) is determined by the values of \( k \) derived from the roots of the equations.