We are asked to graph the function `y=(3x^2+10x-8)/(x^2+4) ` :

Factoring the numerator gives us:

`y=((3x-2)(x+4))/(x^2+4) `

There are no vertical asymptotes. Since the degree of the numerator agrees with the degree of the denominator, the horizontal asymptote is y=3.

The x-intercepts are 2/3 and -4. The y-intercept is -2.

The first derivative is `y'=(-10(x^2-4x-4))/((x^2+4)^2) ` . Using the first derivative test the function decreases for ` x<2-2sqrt(2) ` , has a minimum at `x=2-2sqrt(2) ` , increases on `2-2sqrt(2)<x<2+2sqrt(2) ` with a maximum at `x=2+2sqrt(2) ` , and decreases for `x>2+2sqrt(2) ` .

The graph: