Several comments:

0. You are mixing independent variables there. Are they all x's? Or all t's?

1. Checking to see if your guess works is usually fairly straight-forward. Just plug your solution into the DE and see if it satisfies it.

2. Assuming your RHS is all x's, I would try a particular solution of the form y_p = A x^2 + B x + C + D exp(-3x).

3. Finally, you spell it "differential", not "differencial".