First, note that 3x2-9x+7 has only complex zeroes, so there is no vertical asymptote.
Using calculus the expression needs to be differentiated:
[(3x2-9x+7)(6x+9)-(3x2+9x+17)(6x-9)]/(3x2-9x+7)2=
[18x3-27x2-39x+63-(18x3+27x2+21x-153)]/(3x2-9x+7)2=
(-54x2-60x+216)/(3x2-9x+7)2=0 at max or min.
-54x2-60x+216=0, 9x2+10x-36=0,
x=(-10±√(100+1296))/18=(-10±√1396)/18=(-5±√349)/9, x=1.5202 or -2.6313.
x=-2.6313 is minimum, and x=1.5202 is maximum.
When x=-2.6313 the expression is 0.2738 approx.
When x=-2, the expression is (12-18+17)/(12+18+7)=11/37=0.2973 (approx)>0.2738.
When x=-3, the expression is (27-27+17)/(27+27+7)=17/61=0.2787 (approx)>0.2738, making x=-2.6313 the minimum.
When x=1.5202 the expression is 149.7262 approx.
When x=1 the expression is 29. When x=2 the expression is 47.
When x=1.5202 the expression is 149.7262, much greater than either 29 or 47, making 149.7262 (approx) the maximum.