If `alpha, beta, gamma, delta` are the roots of the equation `x^(4)+Ax^(3)+Bx^(2)+Cx+D=0` such that `alpha beta= gamma delta=k` and A,B,C,D are the roots of `x^(4)-2x^(3)+4x^(2)+6x-21=0` such that `A+B=0` The value of `(alpha+beta)(gamma+delta)` is terms of B and `k` is

Options

A. `B-2k`
B. `B-k`
C. `B+k`
D. `B+2k`

Explanation

To solve the problem step by step, let's break down the information given and derive the required expression.

### Step 1: Understanding the Roots
We have a polynomial equation:
\[ x^4 + Ax^3 + Bx^2 + Cx + D = 0 \]
with roots \( \alpha, \beta, \gamma, \delta \).

From the problem, we know:
\[ \alpha \beta = \gamma \delta = k \]

### Step 2: Using Vieta's Formulas
According to Vieta's formulas for a polynomial of the form \( x^4 + px^3 + qx^2 + rx + s = 0 \):
1. The sum of the roots:
\[ \alpha + \beta + \gamma + \delta = -A \]
2. The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta = B \]
3. The sum of the products of the roots taken three at a time:
\[ \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -C \]
4. The product of the roots:
\[ \alpha\beta\gamma\delta = D \]

### Step 3: Expressing \( \alpha + \beta \) and \( \gamma + \delta \)
Let:
\[ \alpha + \beta = p \]
\[ \gamma + \delta = q \]

From the first equation:
\[ p + q = -A \]

### Step 4: Finding \( \alpha + \beta \) and \( \gamma + \delta \)
We also know:
\[ \alpha\beta + \gamma\delta + (\alpha + \beta)(\gamma + \delta) = B \]
Substituting \( \alpha\beta = k \) and \( \gamma\delta = k \):
\[ k + k + pq = B \]
This simplifies to:
\[ 2k + pq = B \]
Thus, we have:
\[ pq = B - 2k \]

### Step 5: Finding the Value of \( ( \alpha + \beta )( \gamma + \delta ) \)
We need to find \( ( \alpha + \beta )( \gamma + \delta ) \):
\[ ( \alpha + \beta )( \gamma + \delta ) = pq \]
From our previous result:
\[ pq = B - 2k \]

### Final Result
Thus, the value of \( ( \alpha + \beta )( \gamma + \delta ) \) in terms of \( B \) and \( k \) is:
\[ ( \alpha + \beta )( \gamma + \delta ) = B - 2k \]