Answer
$p(x)=-2x^4+4x^3+10x^2-12x$
Work Step by Step
Since $-2,0,1,$ and $3$ are the zeros, the polynomial is in the form of $p(x)=A(x+2)(x-0)(x-1)(x-3)$.
Write in general form:
$p(x)=A(x^2+2x)(x^2-4x+3)$
$p(x)=A(x^4-2x^3-5x^2+6x)$
$p(x)=Ax^4-2Ax^3-5Ax^2+6Ax$
In order that the coefficient of $x^3$ is 4, it must be $-2A=4$ and so $A=-2$.
Now, we have $p(x)=-2x^4+4x^3+10x^2-12x$.
