Form the polynomial equation whose root are
`4+- sqrt(3) ,2 +- i`

Answer

To form the polynomial equation whose roots are \(4 \pm \sqrt{3}\) and \(2 \pm i\), we will follow these steps:

### Step 1: Identify the Roots
The roots given are:
1. \( \alpha = 4 + \sqrt{3} \)
2. \( \beta = 4 - \sqrt{3} \)
3. \( \gamma = 2 + i \)
4. \( \delta = 2 - i \)

### Step 2: Form Factors from Roots
The polynomial can be formed by taking the product of factors corresponding to each root:
\[ (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) = 0 \]
This expands to:
\[ (x - (4 + \sqrt{3}))(x - (4 - \sqrt{3}))(x - (2 + i))(x - (2 - i)) = 0 \]

### Step 3: Simplify the First Pair of Roots
First, we simplify the factors for \( \alpha \) and \( \beta \):
\[ (x - (4 + \sqrt{3}))(x - (4 - \sqrt{3})) = (x - 4 - \sqrt{3})(x - 4 + \sqrt{3}) \]
Using the difference of squares:
\[ = (x - 4)^2 - (\sqrt{3})^2 = (x - 4)^2 - 3 \]
Expanding this:
\[ = x^2 - 8x + 16 - 3 = x^2 - 8x + 13 \]

### Step 4: Simplify the Second Pair of Roots
Now, simplify the factors for \( \gamma \) and \( \delta \):
\[ (x - (2 + i))(x - (2 - i)) = (x - 2 - i)(x - 2 + i) \]
Again using the difference of squares:
\[ = (x - 2)^2 - (i)^2 = (x - 2)^2 - (-1) \]
Expanding this:
\[ = (x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5 \]

### Step 5: Combine the Two Quadratic Factors
Now we have two quadratic factors:
1. \( x^2 - 8x + 13 \)
2. \( x^2 - 4x + 5 \)

We multiply these two quadratics:
\[ (x^2 - 8x + 13)(x^2 - 4x + 5) \]

### Step 6: Expand the Product
Using the distributive property (FOIL method):
\[ = x^2(x^2 - 4x + 5) - 8x(x^2 - 4x + 5) + 13(x^2 - 4x + 5) \]
Calculating each term:
1. \( x^4 - 4x^3 + 5x^2 \)
2. \( -8x^3 + 32x^2 - 40x \)
3. \( 13x^2 - 52x + 65 \)

Combining all these:
\[ x^4 + (-4x^3 - 8x^3) + (5x^2 + 32x^2 + 13x^2) + (-40x - 52x) + 65 \]
This simplifies to:
\[ x^4 - 12x^3 + 50x^2 - 92x + 65 = 0 \]

### Final Polynomial Equation
Thus, the polynomial equation whose roots are \(4 \pm \sqrt{3}\) and \(2 \pm i\) is:
\[ \boxed{x^4 - 12x^3 + 50x^2 - 92x + 65 = 0} \]