Answer
X-intercept(s): $ (\pm3,0),(-1,0)$,
Y-intercept(s): $(0,-9)$,
no symmetry
Work Step by Step
Step 1. For x-intercept(s), let $y=0$, we have $x^2(x+1)-9(x+1)\longrightarrow (x+1)(x^2-9)\longrightarrow (x+1)(x+3)(x-3)\longrightarrow (\pm3,0),(-1,0)$
Step 2. For y-intercept(s), let $x=0$, we have $y=-9\longrightarrow (0,-9)$,
Step 3. To test for x-axis symmetry, replace $(x,y)$ with $(x,-y)$, we have $-y=(x)^3+(x)^2-9(x)-9$ which is different from the original equation, thus it is not symmetric with respect to the x-axis,
Step 4. To test for y-axis symmetry, replace $(x,y)$ with $(-x,y)$, we have $y=(-x)^3+(-x)^2-9(-x)-9$ which is different from the original equation, thus it is not symmetric with respect to the y-axis,
Step 5. To test for origin symmetry, replace $(x,y)$ with $(-x,-y)$, we have $-y=(-x)^3+(-x)^2-9(-x)-9$ which is different from the original equation, thus it is not symmetric with respect to the origin.
