Give an equation with roots 2, 3, 1 - i and 1 + i.
A quartic equation of the form `ax^4 + bx^3 + cx^2 + dx + e = 0` has 4 roots. These can be real, complex, equal or different depending on what the value of a, b, c, d and e is. Complex roots of a quartic equation are always complex conjugate pairs as the coefficients of x are real.
There are an infinite number of equations with roots 2, 3, 1-i and 1+i. The simplest of these is:
(x - 2)(x - 3)(x - 1 + i)(x - 1 - i) = 0
=> (x^2 - 2x - 3x + 6)((x - 1)^2 - i^2) = 0
=> (x^2 - 2x - 3x + 6)(x^2 + 1 - 2x + 1) = 0
=> x^4 - 7x^3 + 18x^2 - 22x + 12 = 0
An equation with roots 2, 3, 1-i and 1+i is x^4 - 7x^3 + 18x^2 - 22x + 12 = 0