Find the polynomial with rational coefficients and whose roots are
`1 +- 2i ,4,2`

Answer

To find the polynomial with rational coefficients whose roots are \(1 \pm 2i\), \(4\), and \(2\), we can follow these steps:

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

### Step 2: Form the Quadratic Polynomial from Complex Roots
The roots \( \alpha \) and \( \beta \) can be used to form a quadratic polynomial. The sum and product of the roots can be calculated as follows:

**Sum of roots:**
\[ \alpha + \beta = (1 + 2i) + (1 - 2i) = 2 \]

**Product of roots:**
\[ \alpha \cdot \beta = (1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - (-4) = 1 + 4 = 5 \]

Thus, the quadratic polynomial with roots \( \alpha \) and \( \beta \) is:
\[ x^2 - (\alpha + \beta)x + \alpha \cdot \beta = x^2 - 2x + 5 \]

### Step 3: Form the Quadratic Polynomial from Real Roots
Next, we form a quadratic polynomial from the real roots \( \gamma \) and \( \delta \):

**Sum of roots:**
\[ \gamma + \delta = 4 + 2 = 6 \]

**Product of roots:**
\[ \gamma \cdot \delta = 4 \cdot 2 = 8 \]

Thus, the quadratic polynomial with roots \( \gamma \) and \( \delta \) is:
\[ x^2 - (\gamma + \delta)x + \gamma \cdot \delta = x^2 - 6x + 8 \]

### Step 4: Multiply the Two Quadratic Polynomials
Now, we need to multiply the two quadratic polynomials obtained:

1. \( x^2 - 2x + 5 \)
2. \( x^2 - 6x + 8 \)

The multiplication is done as follows:
\[ (x^2 - 2x + 5)(x^2 - 6x + 8) \]

### Step 5: Perform the Multiplication
Using the distributive property:
\[ = x^2(x^2 - 6x + 8) - 2x(x^2 - 6x + 8) + 5(x^2 - 6x + 8) \]

Calculating each term:
1. \( x^2(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 \)
2. \( -2x(x^2 - 6x + 8) = -2x^3 + 12x^2 - 16x \)
3. \( 5(x^2 - 6x + 8) = 5x^2 - 30x + 40 \)

Now, combine all these:
\[ x^4 - 6x^3 + 8x^2 - 2x^3 + 12x^2 - 16x + 5x^2 - 30x + 40 \]

### Step 6: Combine Like Terms
Combine the coefficients of like terms:
- \( x^4 \): \( 1 \)
- \( x^3 \): \( -6 - 2 = -8 \)
- \( x^2 \): \( 8 + 12 + 5 = 25 \)
- \( x \): \( -16 - 30 = -46 \)
- Constant: \( 40 \)

Thus, the polynomial is:
\[ x^4 - 8x^3 + 25x^2 - 46x + 40 \]

### Final Answer
The required polynomial is:
\[ \boxed{x^4 - 8x^3 + 25x^2 - 46x + 40} \]