I hate to ask but i am stumped, how do you work out 2e^5x - 7e^2x - 15e^-x=0

Use some variable to take the place of e^x temporarily. Say

u=e^x

=>2u^5-7u^2-15/u=0

=> 2u^6-7u^3-15=0 and u cannot be equal to 0.

use another intermediate variable: m=u^3 (m=e^ (3x))

=>2m^2-7m-15=0

=> (m-5) * (2m+3) = 0

=> m=5 and m=-3/2

=> e^3x=5 and e^3x=-3/2

=> 3x=Ln (5) and 3x=Ln (-3/2) (Ln (-3/2) does not exist so this solution is extraneous)

=> x=Ln (5) / 3