Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively 0,

A + b = 0
ab = √(5)

since b = -a, subst into other equation

ab = √5
a(-a) = √5
-a² = √5
a² = - √5
a =±  (5)^(1/4)·i

so our two zeros are (5)^(1/4) · i and -(5)^(1/4) · i.

Our polynomial has equation



difference of squares: (x + y)(x - y) = x² - y² so



your polynomial is
y = x² + √(5)
y = x^2 + sqrt(5)