It's a bit messy but it can be done. Since you have 2 real zeroes, each with multiplicity of two, you have the form:
f(x) = (x-a)^2 (x-b)^2
So that when you set f(x) = 0, x = a and x = b are your only roots.
Now expand:
(x^2 - 2ax + a^2) (x^2 - 2bx + b^2) = [x^4 - 4x^2 + k]
(The...