How much more than `2x^(2) + 4xy + 2y^(2)` is `5x^(2) + 10xy - y^(2)` ?

Answer

To find out how much more the expression \(5x^2 + 10xy - y^2\) is than the expression \(2x^2 + 4xy + 2y^2\), we need to subtract the first polynomial from the second polynomial. Here’s how to do it step by step:

### Step 1: Write down the expressions
We have two expressions:
- First polynomial: \(2x^2 + 4xy + 2y^2\)
- Second polynomial: \(5x^2 + 10xy - y^2\)

### Step 2: Set up the subtraction
We need to find:
\[ (5x^2 + 10xy - y^2) - (2x^2 + 4xy + 2y^2) \]

### Step 3: Distribute the negative sign
Distributing the negative sign to the first polynomial:
\[ 5x^2 + 10xy - y^2 - 2x^2 - 4xy - 2y^2 \]

### Step 4: Combine like terms
Now, we will combine the like terms:
- For \(x^2\) terms: \(5x^2 - 2x^2 = 3x^2\)
- For \(xy\) terms: \(10xy - 4xy = 6xy\)
- For \(y^2\) terms: \(-y^2 - 2y^2 = -3y^2\)

Putting it all together, we have:
\[ 3x^2 + 6xy - 3y^2 \]

### Step 5: Factor out the common terms
We can factor out a 3 from the expression:
\[ 3(x^2 + 2xy - y^2) \]

### Final Result
Thus, \(5x^2 + 10xy - y^2\) is \(3(x^2 + 2xy - y^2)\) more than \(2x^2 + 4xy + 2y^2\).

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