The differential equation you're working with is:

$y^{(3)} + 2y^{(2)} - 3y^{(1)} - 10y = 8xe^{-2x}$

This is a third-order linear non-homogeneous differential equation. To solve this, we will follow a standard approach that includes two parts:

1. Solving the homogeneous equation:

First, we need to solve the corresponding homogeneous equation:

$y^{(3)} + 2y^{(2)} - 3y^{(1)} - 10y = 0$

This is a linear differential equation with constant coefficients. We solve it by finding the characteristic equation:

$r^3 + 2r^2 - 3r - 10 = 0$

We can solve this cubic equation for the roots $r$, and then the general solution will be a linear combination of terms involving $e^{rx}$, based on the roots we find.

2. Finding a particular solution:

The non-homogeneous part of the equation is $8xe^{-2x}$. We use the method of undetermined coefficients to guess the particular solution.

Since the right-hand side involves $xe^{-2x}$, we will try a particular solution of the form:

$y_p = (Ax^2 + Bx)e^{-2x}$

where $A$ and $B$ are constants to be determined.

3. Combining the solutions:

The general solution will be the sum of the homogeneous solution $y_h$ and the particular solution $y_p$:

$y(x) = y_h + y_p$

Would you like to proceed with solving the characteristic equation and finding the particular solution, or would you like more details on any specific part of the process?