[FONT="]I tried using an online calculator for y''-3y'+2y=2x+4e[/FONT][FONT="]3x[/FONT][FONT="] and the answer they gave was y=c1e[/FONT][FONT="]2x[/FONT][FONT="]+c2e[/FONT][FONT="]x[/FONT][FONT="]+x+1.5+2e[/FONT][FONT="]3x[/FONT][FONT="] .However, isn't the yp part wrong as the answer neglected the fact that there is resonance?[/FONT]

No. The right hand side has nothing to do with the characteristic polynomial.

The roots 1 and 2 give terms in $e^{x}$ and $e^{2x}$ to form the complementary solution. These could only resonate with forcing terms (the right hand side) in $e^{x}$ and $e^{2x}$ respectively.