What should be added to `x^(2) +xy + y^(2)` to obtain `4x^(2) + 8xy`?

Answer

To find what should be added to the expression \( x^2 + xy + y^2 \) to obtain \( 4x^2 + 8xy \), we can follow these steps:

### Step 1: Set Up the Equation
We start with the expression we have and the expression we want to achieve:
\[ x^2 + xy + y^2 + A = 4x^2 + 8xy \]
Here, \( A \) is the expression we need to find.

### Step 2: Rearrange the Equation
To find \( A \), we can rearrange the equation:
\[ A = (4x^2 + 8xy) - (x^2 + xy + y^2) \]

### Step 3: Distribute the Negative Sign
Now, we will distribute the negative sign across the terms in the parentheses:
\[ A = 4x^2 + 8xy - x^2 - xy - y^2 \]

### Step 4: Combine Like Terms
Next, we combine the like terms:
- For \( x^2 \): \( 4x^2 - x^2 = 3x^2 \)
- For \( xy \): \( 8xy - xy = 7xy \)
- For \( y^2 \): The term remains as \(-y^2\)

Putting it all together, we get:
\[ A = 3x^2 + 7xy - y^2 \]

### Final Answer
Thus, the expression that should be added is:
\[ \boxed{3x^2 + 7xy - y^2} \]

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