Question

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Explanation:

Step1: Factor the cubic function

First, we try to factor \( f(x)=x^{3}+6x^{2}+11x + 6 \). We can use the rational root theorem. The possible rational roots are factors of 6, i.e., \( \pm1,\pm2,\pm3,\pm6 \).
Testing \( x=-1 \): \( f(-1)=(-1)^{3}+6(-1)^{2}+11(-1)+6=-1 + 6-11 + 6=0 \). So \( (x + 1) \) is a factor.
We perform polynomial division or use synthetic division. Dividing \( x^{3}+6x^{2}+11x + 6 \) by \( (x + 1) \):
Using synthetic division:

$$
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So the quotient is \( x^{2}+5x + 6 \), which factors further as \( (x + 2)(x + 3) \). Thus, \( f(x)=(x + 1)(x + 2)(x + 3) \).

Step2: Find the x - intercepts

The x - intercepts occur where \( f(x)=0 \). From the factored form \( f(x)=(x + 1)(x + 2)(x + 3) \), we set each factor equal to zero:
\( x+1 = 0\Rightarrow x=-1 \); \( x + 2=0\Rightarrow x=-2 \); \( x+3 = 0\Rightarrow x=-3 \). So the x - intercepts are at \( x=-3,-2,-1 \).

Step3: Analyze the end - behavior

For a cubic function of the form \( f(x)=ax^{3}+bx^{2}+cx + d \), when \( a>0 \) (in our case \( a = 1>0 \)), as \( x
ightarrow+\infty \), \( f(x)
ightarrow+\infty \) and as \( x
ightarrow-\infty \), \( f(x)
ightarrow-\infty \).

Step4: Analyze the behavior between the roots

• For \( x<-3 \), let's take \( x=-4 \). \( f(-4)=(-4 + 1)(-4 + 2)(-4 + 3)=(-3)(-2)(-1)=-6<0 \).
• For \( -3
• For \( -2
• For \( x>-1 \), let's take \( x = 0 \). \( f(0)=(0 + 1)(0 + 2)(0 + 3)=6>0 \).

To identify the graph, we look for a cubic graph that crosses the x - axis at \( x=-3,-2,-1 \), has the correct end - behavior (falling to the left, rising to the right since \( a = 1>0 \)), and the correct sign changes between the roots as we analyzed above.

(Note: Since the actual graphs are not provided here, but if we assume there are multiple graphs to choose from, the correct graph should have x - intercepts at \( x=-3,-2,-1 \), and the shape consistent with the end - behavior and the sign changes between the roots.)

Answer:

(The graph with x - intercepts at \( x=-3,-2,-1 \), falling as \( x
ightarrow-\infty \) and rising as \( x
ightarrow+\infty \), and with the correct sign changes between the roots. If the options are labeled, for example, if Option A has these features, then the answer is A. Option Text (where Option Text describes the graph with x - intercepts at - 3, - 2, - 1 and the correct end - behavior and sign changes).)