Add `x^2+y^2+3xy-6 and 2x^2-4y^2-xy+5`

Answer

To solve the problem of adding the algebraic expressions \(x^2 + y^2 + 3xy - 6\) and \(2x^2 - 4y^2 - xy + 5\), we will follow these steps:

### Step 1: Write down the expressions to be added
We start with the two expressions:
\[ x^2 + y^2 + 3xy - 6 \]
and
\[ 2x^2 - 4y^2 - xy + 5 \]

### Step 2: Combine the expressions
We add the two expressions together:
\[ (x^2 + y^2 + 3xy - 6) + (2x^2 - 4y^2 - xy + 5) \]

### Step 3: Rearrange the terms
Now, we will rearrange the terms by grouping like terms together:
\[ (x^2 + 2x^2) + (y^2 - 4y^2) + (3xy - xy) + (-6 + 5) \]

### Step 4: Simplify each group
Now we simplify each group:
1. For \(x^2 + 2x^2\):
\[ 1x^2 + 2x^2 = 3x^2 \]
2. For \(y^2 - 4y^2\):
\[ 1y^2 - 4y^2 = -3y^2 \]
3. For \(3xy - xy\):
\[ 3xy - 1xy = 2xy \]
4. For \(-6 + 5\):
\[ -6 + 5 = -1 \]

### Step 5: Combine the simplified terms
Now we combine all the simplified terms:
\[ 3x^2 - 3y^2 + 2xy - 1 \]

### Final Answer
Thus, the result of adding the two algebraic expressions is:
\[ 3x^2 - 3y^2 + 2xy - 1 \]

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