The solution is 6te^2t - 6e^2t -5e^t.
Is this correct?
no, it isn't. the "-5" is wrong. try again
you can always check your solution. if you take your solution to be y, then you can find y' and evaluate the expression y' - y and if you get $\displaystyle 6te^{2t}$ you are right, if not, you're not