1(a)

Use the Exponential Shift Theorem with $y = u {\rm e}^x$. We get $((D+1)^2 -2(D+1) + 1) u = D^2 u = 6$. Integrate twice to get $u = 3 x^2 + c_1 x + c_2$. The general solution is $y = (3 x^2 + c_1 x + c_2) {\rm e}^x$.