Solve `(9xy + 7 xy + 9x ^(2)) + ( 8 + x ^(2) - xy - y ^(2) + 1).`

Options

A. `16 xy + 8x ^(2) + xy + y ^(2)+ 9`
B. `16 xy+10 x ^(2)+ xy + y ^(2) + 9`
C. `16 xy + 10 x ^(2) - xy - y ^(2) -9`
D. `16 xy + 10x ^(2) - xy - y ^(2) + 9`

Answer

To solve the expression \((9xy + 7xy + 9x^2) + (8 + x^2 - xy - y^2 + 1)\), we will follow these steps:

### Step 1: Combine like terms
First, we will combine the terms inside the parentheses.

The expression can be rewritten as:
\[ (9xy + 7xy) + (9x^2 + x^2) + (8 + 1) - xy - y^2 \]

### Step 2: Simplify the coefficients
Now, we will simplify the coefficients of like terms:
- For \(xy\): \(9xy + 7xy - xy = (9 + 7 - 1)xy = 15xy\)
- For \(x^2\): \(9x^2 + x^2 = (9 + 1)x^2 = 10x^2\)
- For the constant terms: \(8 + 1 = 9\)

### Step 3: Write the simplified expression
Now, we can write the simplified expression:
\[ 15xy + 10x^2 - y^2 + 9 \]

### Step 4: Final expression
Thus, the final expression after combining and simplifying is:
\[ 15xy + 10x^2 - y^2 + 9 \]

### Step 5: Identify the correct option
From the options given, we can see that the expression can be matched to one of the options provided.

### Final Answer:
The simplified expression is:
\[ 15xy + 10x^2 - y^2 + 9 \]