Question 1099280
Each zero {{{x[k]}}} means the factored form of the polynomial has
as many factors of the form {{{(x - x[k])}}} as the multiplicity of the zero.
That means that the factored form of the polynomial includes
{{{(x-1/2)(x+4)^2(x+I)}}}
For each complex zero, the conjugate complex number is also a zero.
That means that {{{i}}} is also a zero, and that {{{(x-i)}}} also appears as a factor.
So far we have the factors
{{{(x-1/2)(x+4)^2(x+i)(x-i)}}} ,
accounting for the degree 5,
with {{{1}}} for a leading coefficient.
We need to include {{{4}}} as a factor to have 4 for a leading coefficient.
The polynomial is
{{{4(x-1/2)(x+4)^2(x+i)(x-i)=4(x-1/2)(x^2+8x+16)(x^2+1)}}}
={{{(4x-2)(x^4+8x^3+17x^2+8x+16)=
 highlight(4x^5+30x^4+52x^3-2x^2+48x-32)}}} .