Use the quotient rule,

G'(x)=\frac{(x)'(x^2+4)-(x)(x^2+4)'}{(x^2+4)^2}

Thus,

G'(x)=\frac{x^2+4-x(2x)}{(x^2+4)^2}

Thus,

G'(x)=\frac{-x^2+4}{(x^2+4)^2}

Now the second derivative,

G''(x)=\frac{(-x^2+4)'(x^2+4)^2-(-x^2+4)[(x^2+4)^2]'}{(x^2+4)^4}

Thus,

G''(x)=\frac{-2x(x^2+4)^2-4x(-x^2+4)(x^2+4)}{(x^2+4)^4}

Factor,

G''(x)=\frac{-2x(x^2+4)[(x^2+4)+2(-x^2+4)}{(x^2+4)^4}

Thus,

G''(x)=\frac{-2x[x^2+4-2x^2+8]}{(x^2+4)^4}

Thus,

G''(x)=\frac{-2x(-x^2+12)}{(x^2+4)^3}

The inflection points is when G''(x)=0

That is when the numerator is zero.

Thus,

-2x(-x^2+12)=0

x=0

x=+/-sqrt(12)