Expand `: (2x- y + 2)^(3)`

Answer

To expand the expression \((2x - y + 2)^3\), we can use the binomial theorem or the formula for the cube of a trinomial. Here’s a step-by-step solution:

### Step 1: Identify the terms
We can rewrite the expression as:
\[ (2x - y + 2)^3 \]
Let \(a = 2x\), \(b = y\), and \(c = 2\).

### Step 2: Use the formula for the cube of a trinomial
The formula for the cube of a trinomial \((a + b + c)^3\) is given by:
\[ a^3 + b^3 + c^3 + 3(a^2b + a^2c + b^2a + b^2c + c^2a + c^2b) + 6abc \]

### Step 3: Calculate each part
1. **Calculate \(a^3\)**:
\[ (2x)^3 = 8x^3 \]

2. **Calculate \(b^3\)**:
\[ (-y)^3 = -y^3 \]

3. **Calculate \(c^3\)**:
\[ (2)^3 = 8 \]

4. **Calculate \(3(a^2b + a^2c + b^2a + b^2c + c^2a + c^2b)\)**:
- \(3(a^2b) = 3((2x)^2(-y)) = 3(4x^2(-y)) = -12x^2y\)
- \(3(a^2c) = 3((2x)^2(2)) = 3(4x^2(2)) = 24x^2\)
- \(3(b^2a) = 3((-y)^2(2x)) = 3(y^2(2x)) = 6xy^2\)
- \(3(b^2c) = 3((-y)^2(2)) = 3(y^2(2)) = 6y^2\)
- \(3(c^2a) = 3((2)^2(2x)) = 3(4(2x)) = 24x\)
- \(3(c^2b) = 3((2)^2(-y)) = 3(4(-y)) = -12y\)

5. **Calculate \(6abc\)**:
\[ 6(2x)(-y)(2) = -24xy \]

### Step 4: Combine all parts
Now we combine all the calculated parts:
\[ 8x^3 - y^3 + 8 - 12x^2y + 24x^2 + 6xy^2 + 6y^2 + 24x - 12y - 24xy \]

### Final Step: Write the final expanded form
Rearranging the terms, we get:
\[ 8x^3 - y^3 + 24x^2 - 12x^2y + 6xy^2 - 24xy + 6y^2 + 24x - 12y + 8 \]

Thus, the expanded form of \((2x - y + 2)^3\) is:
\[ 8x^3 - y^3 + 24x^2 - 12x^2y + 6xy^2 - 24xy + 6y^2 + 24x - 12y + 8 \]