To find the general solution of the differential equation:

$y''' + y'' + y' + y = e^{-t} + 4t,$

we need to solve both the homogeneous and the non-homogeneous parts.

Step 1: Solve the Homogeneous Part

The homogeneous part of the differential equation is:

$y''' + y'' + y' + y = 0.$

The characteristic equation associated with this differential equation is:

$r^3 + r^2 + r + 1 = 0.$

We will find the roots of this cubic equation to determine the general solution of the homogeneous part.

Step 2: Solve for the Characteristic Roots

The characteristic equation $r^3 + r^2 + r + 1 = 0$ can be factored by trying possible rational roots or using synthetic division.

Let's proceed with solving for $r$.