Need help with this math problem on polynomials?

Question

here's the question:

Find a polynomial with integer coefficients that has the given zeroes: 1+ sqr.root(3i), 2, 2, -1-sqr.root(2)

Answer 1

If the polynomial has real coefficients, complex zeros occur as complex-conjugate pairs. So 1 - i√3 must also be a zero.

If the leading coefficient is 1, then the sum of the zeros is (-1) times the coefficient of the next-highest power of the variable.  Therefore, there must be a zero with irrational part √2 to cancel the irrational part in the last zero listed.

Thus, the factored form of a polynomial with integer coefficients having the given zeros is

(x - (1 + i√3))(x - ((1 - i√3))(x - 2)²(x - (-1 - √2))(x - (-1 + √2))

Observe that

(x - (1 + i√3))(x - ((1 - i√3)) = ((x - 1) - i√3)((x - 1) + i√3)

= (x-1)² - (i√3)²

= x² - 2x + 1 - (-3)

= x² - 2x + 4

and similarly

(x - (-1 - √2))(x - (-1 + √2)) = (x+1)² - 2 = x² + 2x - 1

After multiplying out, this polynomial is

x^6 - 4x^5 + 3x^4 + 14x³ - 48x² + 56x - 16