Answer
$P(x)=-x^4+8x^3 - 25x^2 + 72x -144$.
Work Step by Step
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and if $(x-c)$ is a factor of $f(x)$, then $f(c)=0$.
Because $3i$ is a zero, its conjugate$-3i$ is also a zero.
Hence, here the function is in the form:
$P(x)=a(x-4)^2(x^2+9)=ax^4-8ax^3 +25ax^2 -72a x + 144a$
We know that the $x^2$ coefficient is $-25$. Hence, we can write:
$25a=-25$
$a=-1$
Thus, we have:
$P(x)=-x^4+8x^3 - 25x^2 + 72x -144$.
