If `p & q = p^(2) + 4 pq - q^(2)` then find the value of (3 & 6) + (4 & 5)

Options

A. 98
B. 106
C. 116
D. 126

Answer

To solve the problem, we need to evaluate the expression \( (3 \& 6) + (4 \& 5) \) using the given operation defined as:

\[ p \& q = p^2 + 4pq - q^2 \]

### Step 1: Calculate \( 3 \& 6 \)

Substituting \( p = 3 \) and \( q = 6 \) into the expression:

\[ 3 \& 6 = 3^2 + 4(3)(6) - 6^2 \]

Calculating each term:

- \( 3^2 = 9 \)
- \( 4(3)(6) = 72 \)
- \( 6^2 = 36 \)

Now substituting these values back into the equation:

\[ 3 \& 6 = 9 + 72 - 36 \]

Calculating this gives:

\[ 3 \& 6 = 9 + 72 = 81 \]
\[ 81 - 36 = 45 \]

So, \( 3 \& 6 = 45 \).

### Step 2: Calculate \( 4 \& 5 \)

Now substituting \( p = 4 \) and \( q = 5 \):

\[ 4 \& 5 = 4^2 + 4(4)(5) - 5^2 \]

Calculating each term:

- \( 4^2 = 16 \)
- \( 4(4)(5) = 80 \)
- \( 5^2 = 25 \)

Now substituting these values back into the equation:

\[ 4 \& 5 = 16 + 80 - 25 \]

Calculating this gives:

\[ 4 \& 5 = 16 + 80 = 96 \]
\[ 96 - 25 = 71 \]

So, \( 4 \& 5 = 71 \).

### Step 3: Combine the Results

Now we need to find \( (3 \& 6) + (4 \& 5) \):

\[ (3 \& 6) + (4 \& 5) = 45 + 71 \]

Calculating this gives:

\[ 45 + 71 = 116 \]

### Final Answer

Thus, the final result is:

\[ \boxed{116} \]

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