Expand: `(x-2y+2)^(2)`

Answer

To expand the expression \((x - 2y + 2)^2\), we can use the formula for the square of a trinomial, which is:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]

In our case, we can identify:
- \(a = x\)
- \(b = -2y\)
- \(c = 2\)

Now, we will apply the formula step by step.

### Step 1: Calculate \(a^2\)
\[ a^2 = x^2 \]

### Step 2: Calculate \(b^2\)
\[ b^2 = (-2y)^2 = 4y^2 \]

### Step 3: Calculate \(c^2\)
\[ c^2 = 2^2 = 4 \]

### Step 4: Calculate \(2ab\)
\[ 2ab = 2 \cdot x \cdot (-2y) = -4xy \]

### Step 5: Calculate \(2bc\)
\[ 2bc = 2 \cdot (-2y) \cdot 2 = -8y \]

### Step 6: Calculate \(2ca\)
\[ 2ca = 2 \cdot 2 \cdot x = 4x \]

### Step 7: Combine all the results
Now, we can combine all these results together:
\[ (x - 2y + 2)^2 = x^2 + 4y^2 + 4 - 4xy - 8y + 4x \]

### Final Result
Thus, the expanded form of \((x - 2y + 2)^2\) is:
\[ x^2 + 4y^2 + 4 + 4x - 4xy - 8y \]