Find the slant asymptote, if any, of the rational function. Do not graph the function. (Enter your answers as a comma-separated list of equations. If an answer does not exist, enter DNE.)

f(x) =
x2 − 2x + 15
_____________
x + 7

do a long algebraic or a synthetic division to show that

(x^2 - 2x + 15)/(x+7)
= x-9 + 78/(x+7)

the slant asymptote is
y = x-9

To find the slant asymptote of the rational function f(x) = (x^2 - 2x + 15) / (x + 7), we need to determine if the degree of the numerator is exactly one greater than the degree of the denominator.

In this case, the degree of the numerator is 2 and the degree of the denominator is 1. Therefore, the numerator is one degree higher than the denominator.

To find the slant asymptote, we perform long division between the numerator and the denominator:

x - 9
_________________________
x + 7 | x^2 - 2x + 15

The quotient is x - 9, and there is no remainder. This means that the slant asymptote to the function f(x) is given by the equation y = x - 9.

So, the slant asymptote of the rational function f(x) is y = x - 9.