If `x^(4) +x^(2)y^(2)+y^(4)=21 `, and `x^(2)+xy+y^(2)=3` then what is the value of 4xy?

Options

A. 12
B. 4
C. `-8`
D. `-4`

Answer

To solve the problem, we start with the two equations given:

1. \( x^4 + x^2y^2 + y^4 = 21 \) (Equation 1)
2. \( x^2 + xy + y^2 = 3 \) (Equation 2)

We need to find the value of \( 4xy \).

### Step 1: Use the identity for \( x^4 + y^4 \)

We can use the identity:
\[ x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \]
From Equation 2, we can express \( x^2 + y^2 \) in terms of \( xy \):
\[ x^2 + y^2 = (x^2 + xy + y^2) - xy = 3 - xy \]

### Step 2: Substitute \( x^2 + y^2 \) into the identity

Now, substituting \( x^2 + y^2 \) into the identity:
\[ x^4 + y^4 = (3 - xy)^2 - 2x^2y^2 \]

### Step 3: Expand and simplify

Expanding the right-hand side:
\[ (3 - xy)^2 = 9 - 6xy + x^2y^2 \]
Thus,
\[ x^4 + y^4 = 9 - 6xy + x^2y^2 - 2x^2y^2 = 9 - 6xy - x^2y^2 \]

### Step 4: Substitute back into Equation 1

Now, substituting this back into Equation 1:
\[ 9 - 6xy - x^2y^2 + x^2y^2 = 21 \]
This simplifies to:
\[ 9 - 6xy = 21 \]

### Step 5: Solve for \( xy \)

Rearranging gives:
\[ -6xy = 21 - 9 \]
\[ -6xy = 12 \]
\[ xy = -2 \]

### Step 6: Find \( 4xy \)

Now, we calculate \( 4xy \):
\[ 4xy = 4 \times (-2) = -8 \]

Thus, the value of \( 4xy \) is \( -8 \).

### Final Answer
The value of \( 4xy \) is \( -8 \).

---