What is the slant asymptote of the rational function (2x^3 + 4x^2 + 1)/(x^2 + x + 4) in slope-intercept form?

To find the slant asymptote of the rational function (\frac{2x^3 + 4x^2 + 1}{x^2 + x + 4}), we need to perform polynomial long division since the degree of the numerator (3) is greater than the degree of the denominator (2).

When we divide (2x^3 + 4x^2 + 1) by (x^2 + x + 4), we start by determining how many times the leading term of the denominator (which is (x^2)) goes into the leading term of the numerator (which is (2x^3)). This results in the first term of the quotient being (2x).

Next, we multiply (2x) by the entire denominator (x^2 + x + 4) to calculate the first product:

(2x \cdot (x^2 + x + 4) = 2x^3 + 2x^2 + 8x).

Subtracting this from the original numerator gives:

[ (2x^3 + 4x^2 + 1) - (2x^3