Answer
$G'(x)=\dfrac{2x^{2}+2x+4}{(2x+1)^{2}}$
Work Step by Step
$G(x)=\dfrac{x^{2}-2}{2x+1}$
Differentiate using the quotient rule
$G'(x)=\dfrac{(2x+1)(x^{2}-2)'-(x^{2}-2)(2x+1)'}{(2x+1)^{2}}=\dfrac{(2x+1)(2x)-(x^{2}-2)(2)}{(2x+1)^{2}}=...$
Evaluate the product in the numerator
$...=\dfrac{4x^{2}+2x-2x^{2}+4}{(2x+1)^{2}}=\dfrac{2x^{2}+2x+4}{(2x+1)^{2}}$
$G'(x)=\dfrac{2x^{2}+2x+4}{(2x+1)^{2}}$