x²+14x+9 / x²+2x+9

= 1 + 12x/x²+2x+9

Now , just find maxima and minima of 12x/x²+2x+9

Let f(x)=12x/x²+2x+9

so f'(x) would come out to be = 108-12x²/(x²+2x+9)² .....by quotient rule

since x²+2x+9 is always > 0 the critical points would be only where

108-12x²=0

i.e x = plusminus3

at x = 3;12x/x²+2x+9 = 3/2

at x = -3;12x/x²+2x+9 = -3

So maximum value of  x²+14x+9 / x²+2x+9 = 3/2+1=5/2

& min. value = -3+1=-2