$\left(D^{2}+4 D+3\right) y=8 x e^{x}-6$

Step 1 ：Given the differential equation \(\left(D^{2}+4 D+3\right) y=8 x e^{x}-6\), we need to find the general solution.

Step 2 ：The general solution of a second order linear differential equation with constant coefficients is given by the sum of the complementary function and the particular integral.

Step 3 ：First, we solve the homogeneous equation \(D^{2}y + 4Dy + 3y = 0\) to find the complementary function. The roots of the characteristic equation \(m^2 + 4m + 3 = 0\) will give us the solution of the homogeneous equation. The complementary function is \((C1 + C2*exp(-2*x))*exp(-x)\).

Step 4 ：Next, we find the particular integral. Since the right hand side of the equation is a product of a polynomial and an exponential function, we guess that the particular integral will be of the form \((Ax + B)e^x\).

Step 5 ：We substitute this guess into the original equation and equate coefficients to find the values of A and B. The particular integral solution is \{A: 8/3, B: 0\}.

Step 6 ：Finally, we combine the complementary function and the particular integral to get the general solution of the differential equation. The general solution is \(\frac{8}{3}x e^{x} + (C_{1} + C_{2} e^{-2x}) e^{-x}\).

Step 7 ：\(\boxed{y = \frac{8}{3}x e^{x} + (C_{1} + C_{2} e^{-2x}) e^{-x}}\) is the final answer.