Simplify : `6a^(2)+3ab+5b^(2)-2ab-b^(2)+2a^(2)+4ab+2b^(2)-a^(2)`.

Answer

To simplify the expression \( 6a^2 + 3ab + 5b^2 - 2ab - b^2 + 2a^2 + 4ab + 2b^2 - a^2 \), we will follow these steps:

### Step 1: Group like terms
We will group the terms based on their types: \( a^2 \), \( ab \), and \( b^2 \).

- **Terms with \( a^2 \)**: \( 6a^2 + 2a^2 - a^2 \)
- **Terms with \( ab \)**: \( 3ab - 2ab + 4ab \)
- **Terms with \( b^2 \)**: \( 5b^2 - b^2 + 2b^2 \)

### Step 2: Simplify each group
Now, we will simplify each group separately.

1. **For \( a^2 \)**:
\[ 6a^2 + 2a^2 - a^2 = (6 + 2 - 1)a^2 = 7a^2 \]

2. **For \( ab \)**:
\[ 3ab - 2ab + 4ab = (3 - 2 + 4)ab = 5ab \]

3. **For \( b^2 \)**:
\[ 5b^2 - b^2 + 2b^2 = (5 - 1 + 2)b^2 = 6b^2 \]

### Step 3: Combine the simplified groups
Now, we combine all the simplified groups together:
\[ 7a^2 + 5ab + 6b^2 \]

### Final Answer
The simplified expression is:
\[ \boxed{7a^2 + 5ab + 6b^2} \]

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