Answer :
Use the chain rule. Let
[tex]y=-2(e^{2x}+1)^3[/tex]
and take [tex]u=e^{2x}+1[/tex]. The chain rule says
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm du}\cdot\dfrac{\mathrm du}{\mathrm dx}[/tex]
The relevant derivatives are then
[tex]\dfrac{\mathrm dy}{\mathrm du}=\dfrac{\mathrm d(-2u^3)}{\mathrm du}=-6u^2[/tex]
(power rule)
[tex]\dfrac{\mathrm du}{\mathrm dx}=\dfrac{\mathrm d(e^{2x}+1)}{\mathrm dx}=2e^{2x}[/tex]
(chain rule applied to [tex]e^{2x}[/tex]; the constant vanishes)
So,
[tex]\dfrac{\mathrm dy}{\mathrm dx}=-6u^2\cdot2e^{2x}=-12e^{2x}(e^{2x}+1)^2[/tex]