Expand: `( s + 2)^3`

Options

A. `s^3 + 3s^2 + 12 s + 8`
B. `s^(3) + 3s^(2) + 6s + 8`
C. `s^(3) + 6s^(2) + 12 s + 8`
D. `s^(3) + 6s^(2) + 6s +8`

Answer

To expand the expression \((s + 2)^3\), we can use the binomial theorem or the formula for the cube of a binomial. The formula for the cube of a binomial \((a + b)^3\) is given by:

\[ (a + b)^3 = a^3 + b^3 + 3ab(a + b) \]

In our case, \(a = s\) and \(b = 2\).

### Step 1: Identify \(a\) and \(b\)
Let \(a = s\) and \(b = 2\).

### Step 2: Apply the formula
Using the formula, we can expand \((s + 2)^3\):

\[ (s + 2)^3 = s^3 + 2^3 + 3 \cdot s \cdot 2 \cdot (s + 2) \]

### Step 3: Calculate \(2^3\)
Now, calculate \(2^3\):

\[ 2^3 = 8 \]

### Step 4: Calculate \(3 \cdot s \cdot 2\)
Next, calculate \(3 \cdot s \cdot 2\):

\[ 3 \cdot s \cdot 2 = 6s \]

### Step 5: Substitute back into the equation
Now substitute these values back into the equation:

\[ (s + 2)^3 = s^3 + 8 + 6s(s + 2) \]

### Step 6: Expand \(6s(s + 2)\)
Now, expand \(6s(s + 2)\):

\[ 6s(s + 2) = 6s^2 + 12s \]

### Step 7: Combine all terms
Now, combine all the terms:

\[ (s + 2)^3 = s^3 + 6s^2 + 12s + 8 \]

### Final Answer
Thus, the expanded form of \((s + 2)^3\) is:

\[ s^3 + 6s^2 + 12s + 8 \]