Answer
$f(x)=x^4 - 3 x^3 - 15 x^2 + 19 x + 30
$.
Work Step by Step
If $c$ is a zero of a function with multiplicity $b$ then $(x-c)^b$ is a “factor” of the function.
We are given that the degree is $4$, and the zeros are $-3$, $-1$, $2$ and $5$, hence $f(x)=a(x+3)(x+1)(x-2)(x-5)=a(x^4 - 3 x^3 - 15 x^2 + 19 x + 30
).$
If $a=1$ then the function is $f(x)=x^4 - 3 x^3 - 15 x^2 + 19 x + 30
$.
