If ` p( x + y)^(2) = 5 and q ( x -y )^(2) = 3 ` then the simplified value of ` p^(2) ( x + y)^(2) + 4 pq xy - q^(2) ( x - y) ^(2)` is

Options

A. `- (p + q)`
B. ` 2 ( p + q)`
C. p + q
D. `- 2 ( p + q)`

Answer

To solve the given problem, we have the equations:

1. \( p(x + y)^2 = 5 \)
2. \( q(x - y)^2 = 3 \)

We need to simplify the expression:

\[ p^2 (x + y)^2 + 4pqxy - q^2 (x - y)^2 \]

### Step 1: Express \( (x + y)^2 \) and \( (x - y)^2 \)

From the first equation, we can express \( (x + y)^2 \):

\[ (x + y)^2 = \frac{5}{p} \]

From the second equation, we can express \( (x - y)^2 \):

\[ (x - y)^2 = \frac{3}{q} \]

### Step 2: Substitute into the expression

Now we substitute these values into the expression we need to simplify:

\[ p^2 \left(\frac{5}{p}\right) + 4pqxy - q^2 \left(\frac{3}{q}\right) \]

### Step 3: Simplify each term

1. The first term simplifies to:

\[ p^2 \cdot \frac{5}{p} = 5p \]

2. The second term remains as:

\[ 4pqxy \]

3. The third term simplifies to:

\[ -q^2 \cdot \frac{3}{q} = -3q \]

### Step 4: Combine the terms

Now we can combine all the terms:

\[ 5p + 4pqxy - 3q \]

### Step 5: Substitute values of \( p \) and \( q \)

We know from the original equations that \( p \) and \( q \) can be derived from the values of \( (x+y)^2 \) and \( (x-y)^2 \). Let’s assume values for \( x \) and \( y \) to find \( p \) and \( q \).

Assuming \( x = 1 \) and \( y = 0 \):

1. From \( p(1 + 0)^2 = 5 \), we have \( p = 5 \).
2. From \( q(1 - 0)^2 = 3 \), we have \( q = 3 \).

### Step 6: Substitute \( p \) and \( q \) into the expression

Now substitute \( p = 5 \) and \( q = 3 \):

\[ 5(5) + 4(5)(3)(1)(0) - 3(3) \]

This simplifies to:

\[ 25 + 0 - 9 = 16 \]

### Final Answer

The simplified value of \( p^2 (x + y)^2 + 4pqxy - q^2 (x - y)^2 \) is:

\[ \boxed{16} \]