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 equation \( x^3 + 2x^2 + 2x + 1 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given polynomial is \( x^3 + 2x^2 + 2x + 1 = 0 \). Here, we identify the coefficients:
- \( a_0 = 1 \) (coefficient of \( x^3 \))
- \( a_1 = 2 \) (coefficient of \( x^2 \))

### Step 2: Use Vieta's formulas to find the sum of the roots
According to Vieta's formulas, the sum of the roots \( \alpha_1 + \alpha_2 + \alpha_3 \) of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) is given by:
\[ \text{Sum of roots} = -\frac{a_1}{a_0} \]
Substituting the values we identified:
\[ \text{Sum of roots} = -\frac{2}{1} = -2 \]

### Step 3: Calculate the square of the sum of the roots
Now we need to find the square of the sum of the roots:
\[ (\text{Sum of roots})^2 = (-2)^2 = 4 \]

### Final Answer
Thus, the square of the sum of the roots of the equation \( x^3 + 2x^2 + 2x + 1 = 0 \) is:
\[ \boxed{4} \]

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