The square of sum of the roots of the equation `x^(3)+2x^(2)+2x+1=0` equals to

Options

A. `-2`
B. 2
C. 4
D. `-4`

Answer

To find the square of the sum of the roots of the cubic equation \( x^3 + 2x^2 + 2x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given cubic polynomial is in the form \( ax^3 + bx^2 + cx + d = 0 \). Here, we identify:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 2 \)
- \( d = 1 \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas, the sum of the roots \( \alpha + \beta + \gamma \) of the polynomial can be calculated using the formula:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
Substituting the values of \( b \) and \( a \):
\[ \alpha + \beta + \gamma = -\frac{2}{1} = -2 \]

### Step 3: Calculate the square of the sum of the roots
Now, we need to find the square of the sum of the roots:
\[ (\alpha + \beta + \gamma)^2 = (-2)^2 = 4 \]

### Final Answer
Thus, the square of the sum of the roots of the given equation is \( 4 \).

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