Add the following expressions:
`x^(2)-2xy+3y^(2), 5y^(2)+3xy-6x^(2)`

Answer

To add the expressions \( x^2 - 2xy + 3y^2 \) and \( 5y^2 + 3xy - 6x^2 \), we will follow these steps:

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

### Step 2: Combine the expressions
Next, we combine the two expressions into one:
\[ (x^2 - 2xy + 3y^2) + (5y^2 + 3xy - 6x^2) \]

### Step 3: Rearrange the terms
Now, we rearrange the terms to group like terms together:
\[ x^2 - 6x^2 - 2xy + 3xy + 3y^2 + 5y^2 \]

### Step 4: Combine like terms
Now we will combine the like terms:
- For \( x^2 \) terms: \( x^2 - 6x^2 = -5x^2 \)
- For \( xy \) terms: \( -2xy + 3xy = 1xy \) or simply \( xy \)
- For \( y^2 \) terms: \( 3y^2 + 5y^2 = 8y^2 \)

### Step 5: Write the final expression
Putting it all together, we have:
\[ -5x^2 + xy + 8y^2 \]

### Final Answer:
The sum of the expressions is:
\[ -5x^2 + xy + 8y^2 \]

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