Determine if the function below has a horizontal or a slant asymptote, and explain why. $$ f(x)=\frac{-11 x-23 x^{2}-6-12 x^{3}}{2+x+4 x^{2}} $$ Answer Attempt 1 out of 2 The function has a $$ \quad \vee $$ asymptote because the degree of the numerator is $$ \square $$ the degree of the denominator. As $$ x $$ approaches infinity (positive or negative), the value of $$ y $$ approaches $$ \square $$.

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To determine if the function $$ f(x) = \frac{-11x - 23x^2 - 6 - 12x^3}{2 + x + 4x^2} $$ has a horizontal or slant asymptote, we need to analyze the degrees of the numerator and the denominator.

1. **Identify the degrees of the numerator and denominator:**
   - The numerator is $$ -11x - 23x^2 - 6 - 12x^3 $$. The highest degree term is $$ -12x^3 $$, so the degree of the numerator $$ n = 3 $$.
   - The denominator is $$ 2 + x + 4x^2 $$. The highest degree term is $$ 4x^2 $$, so the degree of the denominator $$ m = 2 $$.

2. **Compare the degrees:**
   - Since the degree of the numerator $$ n = 3 $$ is greater than the degree of the denominator $$ m = 2 $$ (i.e., $$ n > m $$), the function does not have a horizontal asymptote. Instead, it may have a slant (oblique) asymptote.

3. **Finding the slant asymptote:**
   - To find the slant asymptote, we perform polynomial long division of the numerator by the denominator.

Let's perform the polynomial long division to find the slant asymptote.

### Polynomial Long Division

1. Divide the leading term of the numerator by the leading term of the denominator:
   $$
   \frac{-12x^3}{4x^2} = -3x
   $$

2. Multiply the entire denominator by $$-3x$$:
   $$
   -3x(2 + x + 4x^2) = -6x - 3x^2 - 12x^3
   $$

3. Subtract this from the original numerator:
   $$
   (-11x - 23x^2 - 6 - 12x^3) - (-12x^3 - 3x^2 - 6x) = (-11x + 6x) + (-23x^2 + 3x^2) - 6 = -5x - 20x^2 - 6
   $$

4. Now, divide the new leading term by the leading term of the denominator:
   $$
   \frac{-20x^2}{4x^2} = -5
   $$

5. Multiply the entire denominator by $$-5$$:
   $$
   -5(2 + x + 4x^2) = -10 - 5x - 20x^2
   $$

6. Subtract this from the previous result:
   $$
   (-5x - 20x^2 - 6) - (-20x^2 - 5x - 10) = -6 + 10 = 4
   $$

The result of the division is:
$$
f(x) = -3x - 5 + \frac{4}{2 + x + 4x^2}
$$

### Conclusion

As $$ x $$ approaches infinity (positive or negative), the term $$ \frac{4}{2 + x + 4x^2} $$ approaches $$ 0 $$. Therefore, the slant asymptote is given by:
$$
y = -3x - 5
$$

### Final Answer

The function has a slant asymptote because the degree of the numerator is greater than the degree of the denominator. As $$ x $$ approaches infinity (positive or negative), the value of $$ y $$ approaches $$ -3x - 5 $$.
The function has a slant asymptote $$ y = -3x - 5 $$ because the degree of the numerator is greater than the degree of the denominator. As $$ x $$ approaches infinity, $$ y $$ approaches $$ -3x - 5 $$.

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