Answer
$f(x)=-2x^{3}-8x^{2}$
Work Step by Step
If $k$ is a zero, then $(x-k)$ is a factor of $f(x)$ ... (factor theorem)
So,
$(x+4)$ is a factor of f,
$(x-0)$ is a factor of f.
The number of times $(x-k)$ occurs as a factor is referred to as the multiplicity of the zero.So,
$(x+4)$ is a factor of f, degree=1
$(x-0)^{2}=x^{2}$ is a factor of f, degree = 2
(the sum of degrees is 3)
Since f has degree 3, $\quad f(x)=ax^{2}(x+4)$
To find $a,$ use the given information: $f(-1)=-6$
$a(-1)^{2}(-1+4)=-6$
$3a=-6$
$a=-2$
Thus,
$f(x)=-2x^{2}(x+4)$
$f(x)=-2x^{3}-8x^{2}$
