Finding general solution for a nonhomogeneous system of equations

$\{\begin{array}{l}{x}_{1}^{\prime}={x}_{2}+2{e}^{t}\\ {x}_{2}^{\prime}={x}_{1}+{t}^{2}\end{array}$

I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:

$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)={C}_{1}\left(\begin{array}{c}{e}^{t}+{e}^{-t}\\ {e}^{t}-{e}^{-t}\end{array}\right)+{C}_{2}\left(\begin{array}{c}{e}^{t}-{e}^{-t}\\ {e}^{t}+{e}^{-t}\end{array}\right)$

Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants ${C}_{1}$ and ${C}_{2}$ a variable.

$\{\begin{array}{l}{x}_{1}^{\prime}={x}_{2}+2{e}^{t}\\ {x}_{2}^{\prime}={x}_{1}+{t}^{2}\end{array}$

I want to find the general solution for it. I started by finding the general solution for the homogeneous equations:

$\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)={C}_{1}\left(\begin{array}{c}{e}^{t}+{e}^{-t}\\ {e}^{t}-{e}^{-t}\end{array}\right)+{C}_{2}\left(\begin{array}{c}{e}^{t}-{e}^{-t}\\ {e}^{t}+{e}^{-t}\end{array}\right)$

Now I need to find a "specific" solution for the nonhomogeneous equations but I have problems applying the method in which I make constants ${C}_{1}$ and ${C}_{2}$ a variable.