Solve the equation
`x^4 +4x^3 +5x^2 +2x -2=0` given that `-1 + sqrt(-1 )` is a root

Answer

To solve the equation \( x^4 + 4x^3 + 5x^2 + 2x - 2 = 0 \) given that \( -1 + \sqrt{-1} \) (which is \( -1 + i \)) is a root, we can follow these steps:

### Step 1: Identify the roots
Since \( -1 + i \) is a root, its conjugate \( -1 - i \) must also be a root due to the property of complex roots in polynomials with real coefficients.

### Step 2: Form the quadratic factor
The roots \( -1 + i \) and \( -1 - i \) can be used to form a quadratic factor:
\[ (x - (-1 + i))(x - (-1 - i)) = (x + 1 - i)(x + 1 + i) \]
Using the difference of squares:
\[ = (x + 1)^2 - i^2 = (x + 1)^2 - (-1) = (x + 1)^2 + 1 \]
Expanding this:
\[ = (x^2 + 2x + 1 + 1) = x^2 + 2x + 2 \]

### Step 3: Divide the original polynomial
Now we need to divide the original polynomial \( f(x) = x^4 + 4x^3 + 5x^2 + 2x - 2 \) by the quadratic factor \( x^2 + 2x + 2 \).

Using polynomial long division:

1. Divide the leading term \( x^4 \) by \( x^2 \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x^2 + 2x + 2 \):
\[ x^2(x^2 + 2x + 2) = x^4 + 2x^3 + 2x^2 \]
3. Subtract this from the original polynomial:
\[ (x^4 + 4x^3 + 5x^2 + 2x - 2) - (x^4 + 2x^3 + 2x^2) = 2x^3 + 3x^2 + 2x - 2 \]
4. Repeat the process: Divide \( 2x^3 \) by \( x^2 \) to get \( 2x \).
5. Multiply \( 2x \) by \( x^2 + 2x + 2 \):
\[ 2x(x^2 + 2x + 2) = 2x^3 + 4x^2 + 4x \]
6. Subtract:
\[ (2x^3 + 3x^2 + 2x - 2) - (2x^3 + 4x^2 + 4x) = -x^2 - 2x - 2 \]
7. Divide \( -x^2 \) by \( x^2 \) to get \( -1 \).
8. Multiply \( -1 \) by \( x^2 + 2x + 2 \):
\[ -1(x^2 + 2x + 2) = -x^2 - 2x - 2 \]
9. Subtract:
\[ (-x^2 - 2x - 2) - (-x^2 - 2x - 2) = 0 \]

Thus, we have:
\[ f(x) = (x^2 + 2x + 2)(x^2 + 2x - 1) \]

### Step 4: Solve the remaining quadratic
Now we need to solve \( x^2 + 2x - 1 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2} \]

### Final Roots
Thus, the complete set of roots for the equation \( x^4 + 4x^3 + 5x^2 + 2x - 2 = 0 \) are:
1. \( -1 + i \)
2. \( -1 - i \)
3. \( -1 + \sqrt{2} \)
4. \( -1 - \sqrt{2} \)