Add the following expressions:
`3n^(2)+5mn-6m^(2), 2m^(2)-3mn -4n^(2), 2mn-3m^(2)-7n^(2)`

Answer

To add the given algebraic expressions, we will follow these steps:

### Step 1: Write down the expressions
The expressions we need to add are:
1. \(3n^2 + 5mn - 6m^2\)
2. \(2m^2 - 3mn - 4n^2\)
3. \(2mn - 3m^2 - 7n^2\)

### Step 2: Group like terms
We will group the like terms together. The like terms are those that have the same variables raised to the same powers.

- For \(n^2\) terms: \(3n^2\), \(-4n^2\), and \(-7n^2\)
- For \(mn\) terms: \(5mn\), \(-3mn\), and \(2mn\)
- For \(m^2\) terms: \(-6m^2\), \(2m^2\), and \(-3m^2\)

### Step 3: Add the coefficients of like terms
Now we will add the coefficients of each group of like terms:

1. **For \(n^2\) terms:**
\[ 3n^2 - 4n^2 - 7n^2 = (3 - 4 - 7)n^2 = -8n^2 \]

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

3. **For \(m^2\) terms:**
\[ -6m^2 + 2m^2 - 3m^2 = (-6 + 2 - 3)m^2 = -7m^2 \]

### Step 4: Combine the results
Now we combine all the results from the like terms:
\[ -8n^2 + 4mn - 7m^2 \]

### Final Answer
The final result after adding the three expressions is:
\[ -8n^2 + 4mn - 7m^2 \]

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