The sum of `3p^(2)q^(2)- 5pq + 4 " and " 7 + 7pq - 2p^(2)q^(2)` is equal to

Options

A. `p^(2)q^(2) - 2pq + 11`
B. `-p^(2)q^(2) + 2pq +11`
C. `-p^(2) q^(2) -2pq +11`
D. `p^(2)q^(2) + 2pq +11`

Answer

To find the sum of the expressions \(3p^2q^2 - 5pq + 4\) and \(7 + 7pq - 2p^2q^2\), we will follow these steps:

### Step 1: Write down the expressions
We have two expressions:
1. \(3p^2q^2 - 5pq + 4\)
2. \(7 + 7pq - 2p^2q^2\)

### Step 2: Combine the expressions
We will add the two expressions together:
\[ (3p^2q^2 - 5pq + 4) + (7 + 7pq - 2p^2q^2) \]

### Step 3: Rearrange and group like terms
Now, let's rearrange and group the like terms:
\[ (3p^2q^2 - 2p^2q^2) + (-5pq + 7pq) + (4 + 7) \]

### Step 4: Simplify each group
Now, we will simplify each group:
1. For \(p^2q^2\) terms:
\[ 3p^2q^2 - 2p^2q^2 = (3 - 2)p^2q^2 = 1p^2q^2 = p^2q^2 \]
2. For \(pq\) terms:
\[ -5pq + 7pq = (-5 + 7)pq = 2pq \]
3. For the constant terms:
\[ 4 + 7 = 11 \]

### Step 5: Combine the simplified terms
Now, we can combine all the simplified terms:
\[ p^2q^2 + 2pq + 11 \]

### Final Answer
Thus, the sum of the two expressions is:
\[ p^2q^2 + 2pq + 11 \]

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