Assume that y=e⁴ᵗ(At²+Bt+C) where A, B, C are constants to be determined.
-6y=e⁴ᵗ(-6At²-6Bt-6C) and y'=e⁴ᵗ(4At²+4Bt+4C+2At+B)=e⁴ᵗ(4At²+2At+4Bt+B+4C)
4y'=e⁴ᵗ[16At²+8At+16Bt+4B+16C]
y''=e⁴ᵗ[16At²+16t(A+B)+2A+8B+16C]
Add these three equations together to get the original DE:
e⁴ᵗ[26At²+24At+26Bt+2A+12B+26C]=e⁴ᵗ(-8t²+t-4).
Equating coefficients:
t²: 26A=-8, A=-4/13.
t: 24A+26B=1, -96/13+26B=1, 26B=109/13, B=109/338.
Constant: 2A+12B+26C=-4, -8/13+654/169+26C=-4, 26C=-1226/169, C=-613/2197.
Therefore y=e⁴ᵗ(-4t²/13+109t/13-613/2197).