If `x= -2 - sqrt3i`, where `i= sqrt(-1`, find the value of `2x^(4) + 5x^(3) + 7x^(2)-x+ 41`

Answer

To solve the problem, we need to find the value of the expression \(2x^4 + 5x^3 + 7x^2 - x + 41\) given that \(x = -2 - \sqrt{3}i\).

### Step-by-Step Solution:

1. **Identify the value of \(x\)**:
\[ x = -2 - \sqrt{3}i \]

2. **Set up the equation**:
We can rewrite \(x\) in the form of \(x + 2 = -\sqrt{3}i\).

3. **Square both sides**:
\[ (x + 2)^2 = (-\sqrt{3}i)^2 \]
Expanding the left side:
\[ x^2 + 4x + 4 = -3 \]
Rearranging gives:
\[ x^2 + 4x + 7 = 0 \quad \text{(Equation 1)} \]

4. **Use polynomial long division**:
We need to evaluate \(2x^4 + 5x^3 + 7x^2 - x + 41\). We will divide this polynomial by \(x^2 + 4x + 7\).

5. **Divide \(2x^4\) by \(x^2\)**:
The first term of the quotient is:
\[ 2x^2 \]

6. **Multiply and subtract**:
Multiply \(2x^2\) by \(x^2 + 4x + 7\):
\[ 2x^4 + 8x^3 + 14x^2 \]
Subtract from the original polynomial:
\[ (2x^4 + 5x^3 + 7x^2) - (2x^4 + 8x^3 + 14x^2) = -3x^3 - 7x^2 \]

7. **Next term of the quotient**:
Divide \(-3x^3\) by \(x^2\):
\[ -3x \]

8. **Multiply and subtract again**:
Multiply \(-3x\) by \(x^2 + 4x + 7\):
\[ -3x^3 - 12x^2 - 21x \]
Subtract:
\[ (-3x^3 - 7x^2) - (-3x^3 - 12x^2 - 21x) = 5x^2 + 20x \]

9. **Next term of the quotient**:
Divide \(5x^2\) by \(x^2\):
\[ 5 \]

10. **Final multiplication and subtraction**:
Multiply \(5\) by \(x^2 + 4x + 7\):
\[ 5x^2 + 20x + 35 \]
Subtract:
\[ (5x^2 + 20x + 41) - (5x^2 + 20x + 35) = 6 \]

11. **Final result**:
Therefore, we have:
\[ 2x^4 + 5x^3 + 7x^2 - x + 41 = (x^2 + 4x + 7)(2x^2 - 3x + 5) + 6 \]
Thus, the value of the expression is:
\[ \boxed{6} \]