The function f(x) is rational function. To find vertical asymptotes of a rational function, one looks at the polynomial in the denominator. In this case that denominator polynomial is g(x) = x2+3x -10. Vertical asymptotes (if any) will be located at the zeros of g(x). (That is the values of x that satisfy g(x) =0). Finding the zeros of this particular g(x) is easy because it factors:
g(x) = x2 + 3x -10 = (x+5) (x-2)
So the zeros are x = -5 and x = 2.
Next, one must check for a factor of (x +5) or (x-2) in the numerator polynomial. In this case there is not.
Thus there are two vertical asymptotes - located at x = -5 and x = 2. The asymptotes are vertical lines. The equations of these two lines are:
x = -5 and x = 2
If it had not been possible to factor g(x), one would have used the quadratic formula or a calculator method to find the zeros (if any) of g(x). Remember , if a zero is found for the denominator polynomial, one must check to see that the numerator does not also have a zero at the same value of x. If it does, then there is no corresponding vertical asymptote.