Answer
$4x^5+6x^4+4x^3+4x^2-2$
Work Step by Step
The Complex Conjugate zeroes Theorem states that the conjugate of $i$ is also a zero of the polynomial $P(x)$.
Here, we have four zeroes of the polynomial $U(x)$ of degree $5$.
The factorization of $P(x)$ is given as follows:
$U(x)=2(2x-1)(x+i)(x-i)(x+1)^2$
Apply the difference square formula.
$U(x)=2(2x-1)(x^2+1)(x^2+2x+1)=4(2x^3-x^2+2x-1)(x^2+2x+1)$
Hence, $U(x)=2(2x^5+3x^4+2x^3+2x^2-1)=4x^5+6x^4+4x^3+4x^2-2$
