Question

select the zeros of $h(x) = -x^3 - x^2 + 9x + 9$.
identify the graph of the function.

Explanation:

Step1: Set function to 0

$h(x) = -x^3 - x^2 + 9x + 9 = 0$

Step2: Factor by grouping

$-(x^3 + x^2) + 9(x + 1) = 0$
$-x^2(x + 1) + 9(x + 1) = 0$
$(x + 1)(-x^2 + 9) = 0$

Step3: Factor quadratic term

$(x + 1)(3 - x)(3 + x) = 0$

Step4: Solve for x

$x + 1 = 0 \implies x = -1$
$3 - x = 0 \implies x = 3$
$3 + x = 0 \implies x = -3$

Step5: Analyze end behavior

For $h(x) = -x^3 - x^2 + 9x + 9$, leading term is $-x^3$. As $x\to+\infty$, $h(x)\to-\infty$; as $x\to-\infty$, $h(x)\to+\infty$. The graph with zeros at $x=-3, -1, 3$ and matching end behavior is the top-left graph.

Answer:

Zeros: $x=-3$, $x=-1$, $x=3$
Graph: The top-left provided graph