`(x+2)^3+(x-2)^3`

Answer

To simplify the expression \((x+2)^3 + (x-2)^3\), we can use the formula for the sum of cubes. The formula states that:

\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]

In our case, we can let:
- \(A = (x + 2)\)
- \(B = (x - 2)\)

### Step 1: Identify A and B
We have:
\[ A = (x + 2) \quad \text{and} \quad B = (x - 2) \]

### Step 2: Calculate A + B
Now, we calculate \(A + B\):
\[ A + B = (x + 2) + (x - 2) = x + 2 + x - 2 = 2x \]

### Step 3: Calculate A^2, B^2, and AB
Next, we calculate \(A^2\), \(B^2\), and \(AB\):
- \(A^2 = (x + 2)^2 = x^2 + 4x + 4\)
- \(B^2 = (x - 2)^2 = x^2 - 4x + 4\)
- \(AB = (x + 2)(x - 2) = x^2 - 4\)

### Step 4: Substitute into the sum of cubes formula
Now we substitute these values into the formula:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Substituting the values we found:
\[ A^3 + B^3 = (2x)\left((x^2 + 4x + 4) - (x^2 - 4) + (x^2 - 4x + 4)\right) \]

### Step 5: Simplify the expression inside the parentheses
Now simplify the expression inside the parentheses:
\[ = (x^2 + 4x + 4) - (x^2 - 4) + (x^2 - 4x + 4) \]
This simplifies to:
\[ = x^2 + 4x + 4 - x^2 + 4 + x^2 - 4x + 4 \]
Combining like terms:
\[ = (x^2 - x^2 + x^2) + (4x - 4x) + (4 + 4 + 4) = x^2 + 12 \]

### Step 6: Final expression
Now substitute back into the equation:
\[ A^3 + B^3 = 2x(x^2 + 12) \]
This gives us the final answer:
\[ = 2x^3 + 24x \]

### Summary of the steps:
1. Identify \(A\) and \(B\).
2. Calculate \(A + B\).
3. Calculate \(A^2\), \(B^2\), and \(AB\).
4. Substitute into the sum of cubes formula.
5. Simplify the expression inside the parentheses.
6. Write the final expression.