If i is a zero then -i is also a zero. The factors of the polynomial are x+1, x-½, x+i and x-i. Multiplicity of x+1 means that (x+1)² is a factor. A polynomial is therefore a(x+1)²(x-½)(x-i)(x+i). Choose constant a so as to convert x-½ into integer coefficients only, so a=2 is a solution: 2(x-½)=2x-1 factor.
Polynomial is (x+1)²(2x-1)(x²+1)=(x²+2x+1)(2x³+2x-x²-1)=2x⁵+2x³-x⁴-x²+4x⁴+4x²-2x³-2x+2x³+2x-x²-1.
f(x)=2x⁵+3x⁴+2x³+2x²-1.