(3x2+9x+17)/(3x2+9x+7)=1+10/(3x2+9x+7).
When x is very large (positive or negative), the expression approaches 1 because 10/(3x2+9x+7)→0. So 1 is an asymptote.
3x2+9x+7 is never zero for real x, but the maximum value of the expression is when 10/(3x2+9x+7) is maximum: when the derivative is zero:
-10(3x2+9x+7)-2)(6x+9)=0, so 6x=-9, x=-3/2 or -1.5.
10/(3x2+9x+7)=10/(6.75-13.5+7)=10/0.25=40. So the maximum is 40+1=41.