There is an obvious factor of x², so we can reduce this to the problem of finding zeros of a quadratic by factoring that out. Then, we can determine the zeros of the quadratic factor to complete the finding of zeros for the function.
A factor of x² can be factored out, leaving ...
... f(x) = x²(x² +x +4)
The latter factor has only complex zeros. It can be rewritten by completing the square.
... x² +x +4 = (x² +x +1/4) +(3 3/4) = (x +1/2)² +(3 3/4)
So, the real zeros are where x² = 0, at x = 0. The complex zeros are where the above expression is zero, ...
... (x +0.5)² +3.75 = 0
... (x +0.5)² = -3.75
... x + 0.5 = √-3.75 = ±0.5i√15
... x = -0.5(1 ±i√15)
The function zeros are 0 (multiplicity 2) and -0.5±0.5i√15.
