To determine the slant asymptote of the rational function

$h(x) = \frac{4x^4 + 12x^3 + 4x^2 - x + 3}{4x^3 + 1},$

we need to perform polynomial long division because the degree of the numerator ($4$) is higher than the degree of the denominator ($3$).

Step-by-Step Solution:

1. Set up the division: Divide the numerator ($4x^4 + 12x^3 + 4x^2 - x + 3$) by the denominator ($4x^3 + 1$).

2. First term of the quotient: Divide the leading terms: $\frac{4x^4}{4x^3} = x.$ Multiply $x$ by $4x^3 + 1$, giving: $x(4x^3 + 1) = 4x^4 + x.$ Subtract this from the numerator: $(4x^4 + 12x^3 + 4x^2 - x + 3) - (4x^4 + x) = 12x^3 + 4x^2 - 2x + 3.$

3. Second term of the quotient: Divide the leading terms: $\frac{12x^3}{4x^3} = 3.$ Multiply $3$ by $4x^3 + 1$, giving: $3(4x^3 + 1) = 12x^3 + 3.$ Subtract this from the result: $(12x^3 + 4x^2 - 2x + 3) - (12x^3 + 3) = 4x^2 - 2x.$

4. Remainder ignored for slant asymptote: The quotient is: $y = x + 3.$

The slant asymptote is $y = x + 3$.

Correct Answer:

$\boxed{y = x + 3}$

Would you like a detailed explanation of the polynomial long division process? Here are related questions for further practice:

1. How do you determine slant asymptotes for rational functions in general?
2. What happens if the degree of the numerator is equal to or less than the denominator?
3. Can a rational function have both a horizontal and slant asymptote?
4. How does the remainder from division affect the graph of the function?
5. What is the relationship between polynomial long division and the remainder theorem?

Tip: The slant asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator!